TEACHING GUIDE
Module 1: Special products and Factors
A.
Learning Outcomes
1.
Grade Level Standard
The learner demonstrates understanding of key concepts and principles of algebra, geometry, probability and
statistics as applied, using appropriate technology, in critical thinking, problem solving, reasoning, communicating,
making connections, representations, and decisions in real life.
2.
Content and Performance Standards
Content Standards:
The learner demonstrates understanding of the key concepts of special products and factors of polynomials.
Performance Standards:
The learner is able to formulate real-life problems involving special products and factors and solves these with
utmost accuracy using a variety of strategies.
1
UNPACKING THE STANDARDS FOR UNDERSTANDING
SUBJECT:
Grade 8 Mathematics
QUARTER:
First Quarter
STRAND:
Algebra
TOPIC:
Special Products and Factors
LESSONS:
1. Special Products
2.Factoring
LEARNING COMPETENCIES
a. Identify polynomials which are special products: polynomials with common monomial
factors, trinomials that are product of two binomials, trinomials that are product of squares
of a binomial and products of sum and difference of two terms
b. Find special products and factors of certain polynomials: product of two binomials,
product of a sum and difference of two terms, square of a binomial, cube of a binomial
and product of special case of multiplying a binomial with a trinomial
c. Factor completely different types of polynomials (polynomials with common monomial
factors, difference of two squares, sum and difference of two cubes, perfect square
trinomials, general trinomials) using special formulas, grouping and other techniques.
d. Solve problems involving polynomials and their products and factors.
ESSENTIAL UNDERSTANDING:
ESSENTIAL QUESTION:
Students will understand that unknown How can unknown quantities in geometric
quantities in geometric problems can be problems be solved?
determined by using patterns of special
products and factors.
TRANSFER GOAL:
Apply the concepts of special products and factors to model various real-life situations and
solve related problems.
B.
Planning for Assessment
Product/Performance
The following are products and performances that students are expected to come up within this module.
a. Punnet square containing the desired genes using the concepts of special products.
b. Pictures and / or sketch of a figure that makes use of special products.
c.
Cylindrical containers as trash can model which uses the idea of factoring.
d. Rectangular prism that can be used as packaging box which will demonstrate students’ understanding of special
products and factoring.
2
Assessment Map
TYPE
KNOWLEDGE
PROCESS/SKILLS
UNDERSTANDING
PERFORMANCE
PRE – TEST
Background Knowledge
(Interpretation, Explanation)
Pre-Assessment/
Diagnostic
Gallery Walk
(Interpretation, explanation,
Self – knowledge)
Knowledge Inventory
(Self – knowledge)
IRF Worksheet
(Interpretation, Explanation)
Written Exercises / Drills
(Interpretation, Explanation)
Formative
Quiz
(Interpretation, Explanation)
IRF Worksheet
(Explanation, Self –
knowledge)
Flash Card Drill
(Interpretation,
Self – knowledge)
We have! (Oral
Questioning)
(Interpretation)
Decision Making (Written
exercises)
(Interpretation, Explanation,
Application, Perspective,
Empathy)
Pattern finding in real world
(Application, explanation,
interpretation)
Debate
(Interpretation, explanation,
Application, Empathy, Self –
knowledge, perspective)
Graphic Organizer
(Self – knowledge,
Explanation, interpretation)
IRF Worksheet
(Interpretation, Explanation)
3
Unit Test
(Interpretation, Explanation,
Self – knowledge,
Application)
Packaging activity
(Self – knowledge,
Interpretation, Application,
Explanation)
Misconception checking
(Spotting Errors)
(Self – knowledge,
Explanation, Empathy)
Summative
Summative Test
(Interpretation, Application,
Self – knowledge, Empathy)
3 – 2 – 1 Chart
(Explanation, Application,
Self – knowledge,
Perspective)
Self-Assessment
Muddiest point
Self – knowledge,
Explanation, Perspective
Journal Writing
(Self – knowledge,
Explanation, Perspective)
4
Assessment Matrix (Summative Test)
Levels of Assessment
Knowledge 15%
Process/Skills 25%
Understanding 30%
What will I assess?
How will I assess?
• Identifying polynomials which are special
products: polynomials with common monomial
factors, trinomials that are product of two
binomials, trinomials that are product of squares
of a binomial and products of sum and difference
of two terms.
• finding special products and factors of certain
polynomials: product of two binomials, product
of a sum and difference of two terms, square
of a binomial, cube of a binomial and product
of special case of multiplying a binomial with a
trinomial
• factor completely different types of polynomials
(polynomials with common monomial factors, a
difference of two squares, sum and difference
of two cubes, perfect square trinomials, general
trinomials) using special formulas, grouping and
other techniques.
Paper and pen Test (refer to attached
post – test)
How Will I Score?
1 point for every correct response
Items 1, 2 & 3
Paper and pen Test
(refer to attached post – test)
1 point for every correct response
Items 4, 5, 6, 7 & 8
• Students will understand that unknown quantities
Paper and pen Test (refer to attached
in geometric problems can be determined by
post – test)
using patterns of special products and factors.
• Misconceptions regarding special product and
Items 9, 10, 11, 12, 13 & 14
factors.
1 point for every correct response
5
GRASPS
Paper and pen Test (refer to attached
post – test)
Solve real – life problems involving polynomials and
their products and factors.
Items 15, 16, 17, 18, 19 & 20
Product/
Performance 30%
C.
The learner is able to formulate real-life problems TRANSFER TASK IN GRASPS
involving special products and factors and solves FORM
these with utmost accuracy using a variety of
strategies.
The RER packaging company is
in search for the best packaging for
a new dairy product that they will
introduce to the market. You are a
member of the design department
of RER Packaging Company. Your
company is tapped to create the best
packaging box that will contain two
identical cylindrical containers with
the box’s volume set at 100 in3. The
box has an open top. The cover will
just be designed in reference to the
box’s dimensions. You are to present
the design proposal for the box and
cylinder to the Chief Executive Officer
of the dairy company and head of the
RER Packaging department.
1 point for every correct response
Rubric on packaging box.
Criteria:
1.
2.
3.
4.
Explanation of the proposal
Accuracy of computations
Utilization of the resources
Appropriateness of the model
Planning for Teaching-Learning
Introduction:
This module covers key concepts in special products and factors of polynomials. It is divided into two lessons namely:
(1) Special products, and (2) Factoring. In lesson 1, students will identify polynomials which are special products, find the
product of polynomials using patterns, solve real – life problems involving special products and identify patterns in real – life
which involves special products. In lesson 2, students will factor polynomials completely using the different rules and techniques in factoring, including patterns and concepts on special products. They will also apply their knowledge in factoring to
solve some real – life problems.
6
In all lessons, students are given the opportunity to use their prior knowledge and skills in multiplying and dividing polynomials. Activities are also given to process their knowledge and skills acquired, deepen and transfer their understanding of
the different lessons.
As an introduction to the module, ask the students following questions:
Have you at a certain time asked yourself how a basketball court was painted using less paint? Or how the architect was able to maximize the space of a
building and yet was able to place what the owners want? Or how a carpenter was able to create a utility box using minimal materials? Or how some students
were able to multiply some polynomial expressions in a least number of time?
Allow the students to give their response and process them after. Emphasize to the students their goal after completing this
module and the lessons.
I.
PRE – ASSESSMENT
1. Which mathematical statement is correct?
a.
(2x – y)(3x – y) = 6x2 – 5x2y2 +y2
b. (4x – 5)(4x – 5)= 16x2 + 25
c. (3x – 4)(2x + 7) = 6x2 + 13x – 28
d. (2x + 5)2 = 4x2 + 25
Answer: C
7
2.
Which of the following DOES NOT belong to the group?
1
4
a. c. 1.6(x – 1)2 – 49
4 x – 1 b. x2 – 0.0001y4d. (x + 1)4 – 4x6
Answer: C
3. Which of the following factors gives a product of x2 + 5x + 4?
a. (x + 1)(x + 4)
c. (x + 5)(x – 1)
b. (x + 2)(x + 2)
d. (x + 2)2
Answer: A
4.
A polynomial expression is evaluated for the x- and y-values shown in the table below. Which expression was evaluated
to give the values shown in the third column?
X
Y
Value of the Expression
0
0
0
-1
-1
0
1
1
0
1
-1
4
a. x2 – y2c. x2 – 2xy + y2
2
2
b. x + 2xy + y d.
x3 – y3
Answer: C
5. Find the missing term: (x + ___)(3x + ___) = 3x2 + 27x + 24
a. 6, 4c. 8, 3
b. 4, 6d. 12, 2
Answer C
8
6.
The length of a box is five inches less than twice the width. The height is 4 inches more than three times the width. The
box has a volume of 520 cubic inches. Which of the following equations can be used to find the height of the box?
a. W(2L – 5) (3H + 4) = 520
c. W(2W – 5) (3W – 4) = 520
b. W(2W + 5) (3W – 4) = 520
d. W(2W – 5) (3W + 4) = 520
Answer: D
7. One of the factors of 2a2 + 5a – 12 is a + 4. What is the other factor?
a. 2a – 3
b. 2a + 3
c. 2a – 8
Answer: A
d. 2a + 8
8. The area of a square is 4x2 + 12x + 9 square units. Which expression represents the length of the side?
a. (3x + 2) units
c. (4x + 9) units
b. (2x + 3) units
d. (4x + 3) units
Answer: B
The side of a square is x cm long. The length of a rectangle is 5 cm. longer than the side of the square and the width is
5 cm shorter. Which statement is true?
a. the area of the square is greater than the area of the rectangle
b. the area of the square is less than the area of the rectangle
c. the area of the square is equal to the area of the rectangle
d. the relationship cannot be determined from the given information
Answer: A
9.
10. A square piece of land was rewarded by a master to his servant. They agreed that a portion of it represented by the
rectangle inside, should be used to construct a grotto. How large is the area of the land that is available for the other
uses?
2
a.
4x2 – 9 5 - 2x
b.
4x2 + x + 9 Answer: C
2x - 1
c. 4x2 – 8x – 9 d. 4x2 + 9
2x - 1
9
11. Which value for x will make the largest area of the square with a side of 3x + 2?
3
1
a. - c. - 3
4
0.4d. -0.15
b. Answer: C
12. Which procedure could not be used to solve for the area of the figure below?
1
a. A = 2x (2x + 6) +
(2x)(x + 8)
1
2
c.
A = [2x(2x + 6) + (x + 8)(2x)] – 2( )(x)(x + 8)
2
2
2
A = 4x + 12x + x + 8x
2
2
2
A = [4x + 12x) + (2x + 16x) – (x + 8x)
A = 5x2 + 20x
2x + 6 A = 6x2 + 28x – x2 – 8x
1
b. A = 2x(3x + 14) – 2( )(x)(x + 8)
A = 5x2 + 20x
2x
2
2
2
A = 6x + 28x – x – 8x
1
d. A = 2x(2x + 6)+( )(2 + x)(x + 8)
2
2
A = 5x + 20x
Answer: D
2
A = 4x + 12x + x2 + 8x
13.
x + 8 A = 5x2 + 20x
Your classmate was asked to square (2x – 3), he answered 4x2 – 9. Is his answer correct?
a. Yes, because squaring a binomial always produces a binomial product.
b. Yes, because product rule is correctly applied.
c.
No, because squaring a binomial always produces a trinomial product.
d. No, because the answer must be 4x2 + 9.
Answer: C
10
14. Expression A: 4x2 – 81
Expression B: (2x – 9)(2x + 9)
If x = 2, which statement is true about the given expressions?
a.
A>B
b.
A<B
c.
A=B
d.
A≠B
Answer: C
15. Your sister plans to remodel her closet. She hired a carpenter to do the task. What should your sister do so that the carpenter
can accomplish the task according to what she wants?
a. Show a replica of a closet.
b. Download a picture from the internet.
c. Leave everything to the carpenter.
d. Provide the lay out drawn to scale.
Answer: D
16.
Which of the following standards would best apply in checking the carpenter’s work in item number 15?
a. accuracy of measurements and wise utilization of materials
b. accuracy of measurements and workmanship c. workmanship and artistic design
d. workmanship and wise utilization of materials
Answer: B
17. The city mayor asked you to prepare a floor plan of the proposed day care center in your barangay. The center must have a
small recreational corner. As head of the city engineering office, what will you consider in preparing the plan?
a. Feasibility and budget.
b. Design and budget
Answer: A
c.
d.
Design and Feasibility
Budget and lot area
11
18.
Suppose there is a harvest shortage in your farm. What will you do to ensure a bountiful harvest in your farmland?
a. Hire lot of workers to spread fertilizers in the farmland.
b. Buy numerous sacks of fertilizers and spread it in his farmland.
d. Find the area of the farmland and buy proportionate number of fertilizers.
c.
Solve for the number of fertilizers proportionate to the number of workers.
Answer: C
19.The Punong Barangay in your place noticed that garbage is not properly disposed because of the small bins. As the chairman
of health committee, you were tasked to prepare a garbage bins which can hold 24 ft3 of garbage. However, the spot where
the garbage bins will be placed is limited, how will you maximize the area?
a. Find the dimensions of the planned bin according to the capacity given.
b. Make a trial and error bins until the desired volume is achieved
c.
Solve for the factors of the volume and use it in creating bins.
d. Find the area of the location of the bins
Answer: A
20. As head of the marketing department of a certain construction firm, you are tasked to create a new packaging box for the soap
products. What criteria will you consider in creating the box?
a. Appropriateness and the resources used.
b. Resources used and uniqueness
c.
Appropriateness and uniqueness
d. Appropriateness and capacity
Answer: D
12
LESSON 1
SPECIAL PRODUCTS
Lesson
What to
to Know
Know
What
Let us start our study of this module by reviewing first the concepts on
multiplying polynomials, which is one of the skills needed in the study of this
module. Discuss the questions below with a partner.
1
Special
Products
What to
to Know
Know
What
97 × 103 =
25 × 25 =
99 × 99 × 99 =
Allow the students to answer the following process questions:
Let us start our study of this module by reviewing first the concepts on multiplying
polynomials, which is one of the skills needed in the study of this module. Discuss the
questions below with a partner.
PATTERNS WHERE ARE YOU?
1. What do you notice about the given expressions?
2. Did you solve them easily? Did you notice some patterns in solving their
answers?
3. What technique/s did you use? What difficulties did you encounter?
Have you ever looked around and recognized different patterns? Have you asked
yourself what the world’s environment would look like if there were no patterns? Why do
you think our Creator includes patterns around us?
Look at the pictures below and identify the different patterns on each picture. Discuss
these with a partner and see whether you observe the same pattern.
You can present the following solution to the students:
97 x 103 = (100 – 3)(100 + 3)
25 x 25 = (20 + 5)(20 + 5)
= 1002 – 32
= 202 + 2(20)(5) + 52
= 10000 – 9
= 400 + 200 + 25
= 9991
= 625
99 x 99 x 99 = (100 – 1)3
= 1003 + 3(100)2(- 1) + 3(100)(-1)2 + (-1)3
= 10000000 – 30000 + 300 – 1
= 970 299
http://meganvanderpoel.blogspot. http://gointothestory.blcklst.com/2012/02/
com/2012/09/pattern-precedents. doodling-in-math-spirals-fibonacci-andhtml
being-a-plant-1-of-3.html
Have you ever used patterns in simplifying mathematical expressions? What
advantages have you gained in doing such? Let us see how patterns are used to simplify
mathematical expressions by doing the activity below. Try to multiply the following
numerical expressions. Can you solve the following numerical expressions mentally?
Discuss the given solution to the students and give more numerical
problems and allow students to present their solutions, challenge them to look
for another pattern to solve problems presented. Do this mentally, e.g. (42)
(38), (57)(63), (42)(42).
13
97 × 103 =
25 × 25 =
99 × 99 × 99 =
Now, answer the following questions:
1.
2.
3.
The indicated products can be solved easily using different patterns.
What do you notice about the given expressions?
Did you solve them easily? Did you notice some patterns in finding their answers?
What technique/s did you use? What difficulties did you encounter?
Post the topical questions and allow the students to write their answer on the
Initial portion of the IRF worksheets in their journal notebook, this will enable
you to know if they have idea on the lesson. (1) What makes a product special?
and (2) What patterns are involved in multiplying expression?
The next activity is provided to review the multiplication skills of the students
in polynomials, which is an important skill in completing this module. Allow
the students to do the activity by pair. (Note: If most of the students have not
yet attained the level of mastery in this skill, bridge on the topic multiplying
polynomials)
Are your solutions different from your classmates? What was used in order to find the
products easily?
The problems you have answered are examples of the many situations where
we can apply knowledge of special products. In this lesson, you will do varied activities
which will help you answer the question, “How can unknown quantities in geometric
problems be solved?”
Let’s begin by answering the “I” portion of the IRF Worksheet shown below. Fill it up
by writing your initial answer to the topical focus question:
A ctivity 1
Answers Key
Activity 3
IRF WORKSHEET
Description: Direction: Below is the IRF worksheet which will determine your prior knowledge
about the topical question.
Answer the topical questions: (1) What makes a product special? and
(2) What patterns are involved in multiplying algebraic expressions?
Write your answer in the initial part of the IRF worksheet.
IRF Worksheet
Initial Answer
Revised Answer
Final Answer
A ctivity 2
COMPLETE ME!
Description: Teacher’s Note and Reminders
Don’t
Forget!
Directions: This activity will help you review multiplication of polynomials, the prerequisite skill to complete this module.
Complete the crossword polynomial by finding the indicated products
below. After completing the puzzle, discuss with a partner the questions
1
2
3
that follow.
AcrossDown
1.(a + 3)(a + 3)
1. (a + 9)(a – 9)
4. (b + 4a)22.
(3 + a + b)2
5. 2a ( -8a + 3a) 3.(3b – 4a)(3b – 4a)
6. (b – 2)(b – 4)
5. (-4a + b)(4a + b)
9. -2a(b + 3a – 2)
7. (2 – a)(4 – a)
11.(5b2 + 7a2)(-5b2 + 7a2) 8. (4a3 – 5b2)(4a3 + 5b2)
12.(a – 6b)(a + 6b)
10. (2a + 6b)(2a – 6b)
14
4
5
6
8
11
7
9
10
12
CASE 1:
QU
ES TI O
A ctivity 3
CASE 2:
(x + 5)(x – 5) = x – 25
(a – b)(a + b) = a2 – b2
(x + y)(x – y) = x2 – y2
(x – 8)(x + 8) = x2 – 64
(2x + 5)(2x – 5) = 4x2 – 25
2
CASE 3:
(x + 5)3 = x3 + 15x2 + 75x + 125
(a – b)(a – b)(a – b) = a3 – 3a2b + 3ab2 – b3
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(x + 4)(x + 4)(x + 4) = x3 + 12x2 + 48x + 64
(x + 2y)3 = x3 + 3x2y + 12xy2 + 8y3
?
(x + 5)(x + 5) = x2 + 10x + 25
(a – b)2 = a2 – 2ab + b2
(x + y)(x + y) = x2 + 2xy + y2
(x – 8)2 = 16x + 64
(2x + 5)(2x + 5) = 4x2 – 20x + 25
1.
2.
3.
NS
Activity 3 is provided to reinforce activity 2, but this time students should see
the different patterns and initially will have an idea on the different special
products. Provide station for each group where they will post their outputs.
Allow the students to roam around and observe the different answers of the
other groups. (Note: if you are handling a big class, cases may be given to
more than 1 group to accommodate the class)
GALLERY WALK
Description: Direction: This activity will enable you to review multiplication of polynomials.
Find the indicated product of the expressions that will be handed to your
group. Post your answers on your group station. Your teacher will give
you time to walk around the classroom and observe the answers of the
other groups. Answer the questions that follow.
CASE 4:
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2xz + 2yz
(m + 2n – 3f)2 = m2 + 4n2 + 9f2 + 4mn – 6fm – 12fn
You can process their answers after the activity; allow their outputs to be
posted on their stations all throughout the lesson, so that they can revisit it
on the course of the discussion.
CASE 1:
CASE 2:
(x + 5)(x – 5) =
(a – b)(a + b) =
(x + y)(x – y) =
(x – 8)(x + 8) =
(2x + 5)(2x – 5) =
(x + 5)(x + 5) =
(a – b)2 =
(x + y)(x + y) =
(x – 8)2 =
(2x + 5)(2x + 5) =
CASE 3:
CASE 4:
(x + 5) =
(a – b)(a – b)(a – b) =
(x + y)3 =
(x + 4)(x + 4)(x + 4) =
(x + 2y)3 =
(a + b + c)(a + b + c) =
(x + y + z)(x + y + z) =
(m + 2n – 3f)2 =
3
Teacher’s Note and Reminders
QU
?
NS
ES TI O
Don’t
Forget!
How did you find each indicated product?
Did you encounter any difficulty in finding the products? Why?
What concept did you apply in finding the product?
1.
2.
3.
4.
15
How many terms do the products contain?
Compare the product with its factors. What is the relationship
between the factors and the terms of their product?
Do you see any pattern in the product?
How did this pattern help you in finding the product?
You just tried finding the indicated products through the use of patterns. Are the
techniques applicable to all multiplication problems? When is it applicable and when is it
not?
What to
to Process
Process
What
These are the enabling activities / experiences that the learner will have to
go through to validate their observations in the previous section. Interactive
activities are provided for the students to check their understanding on the
lesson.
Let us now find out what the answer is by doing the next part. What you will learn
in the next sections will enable you to do the final project which involves making of a
packaging box using the concepts of special products and factoring.
Let us start by doing the next activity.
What to
to Process
Process
What
Before performing activity 4 give first a short introduction on what is a square
of binomial and how it is written mathematically. Ask them how they simplify
such expressions.
Activity 4 can be performed by pair or as a group. Roam around to observe
whether the students are doing the activity correctly. Use process questions
to guide the students.
Teacher’s Note and Reminders
Your goal in this section is to learn and understand key concepts related to
finding special products. There are special forms of algebraic expressions whose
products are readily seen and these are called special products. There are certain
conditions which would make a polynomial special. Discovering these conditions will
help you find the product of algebraic expressions easily. Let us start in squaring a
binomial
The square of a binomial which is expressed as (x + y)2 or (x + y)(x + y) and(x – y)2 or
(x – y)(x – y) respectively. In your previous grade you did this by applying the FOIL method,
which is sometimes tedious to do. There is an easier way in finding the desired product and
that is what are we going to consider here.
A ctivity 4 FOLD TO SQUARE
Description: Directions: Don’t
o
F rget!
In this activity, you will model square of a binomial through paper
folding. Investigate the pattern that can be produced in this activity. This
pattern will help you find the square of a binomial easily. You can do this
individually or with a partner.
Get a square paper measuring 8” × 8”
1. Fold the square paper 1” to an edge and make a crease.
2. Fold the upper right corner by 1” and make a crease.
3. Unfold the paper.
4. Continue the activity by creating another model for squaring a
binomial by changing the measures of the folds to 2 in. and 3 in.
Then answer the questions below.
7
7
1
16
x
1
x
y
y
FIRST
TERM
SECOND
TERM
LAST
TERM
(x + 1)2
x
2
2x
1
(x + 2)2
x2
4x
4
2.
3.
(x + 3)2
x2
6x
9
4.
(x + y)2
x2
2xy
y2
?
1.
NS
QU
ES TI O
5.
Let them complete the table and emphasize that the first terms are the area
of big squares, second terms are the total areas of the rectangles and the
last terms are the areas of the small squares.
How many different regions are formed? What geometric figures
are formed? Give the dimensions of each region?
What is the area of each region?
If the longer part is represented by x, what will be its area? by x and
1?
What is the sum of the areas? Write the sum of areas in the box
below.
If 1 is replaced by y, what will be the area?
FIRST TERM
SECOND TERM
LAST TERM
(x + 1)2
(x + 2)2
(x + 3)2
(x + y)2
Note: Use process questions to guide the students in completing the table
and recognized the pattern that exists in squaring binomials. Provide
opportunity to the students to create their rule in this special product.
After completing the activity, process their answers and lead them in
the discovery of the rule. Give more examples to the students to firm their
understanding of the lesson. You can use video lessons, if available, in the
discussion of this topic. URL’s are provided in the students learning modules.
Challenge the students to ponder on the equation (a + b)2 = a2 + b2. Let
the students realize that the two expressions are not equal and that the
product of squaring a binomial is a perfect square trinomial.
Did you find any pattern? What pattern is it?
Teacher’s Note and Reminders
1.
2.
3.
How is the first term of the product related to the first term of the given binomial?
How is the last term of the product related to the last term of the given binomial?
What observation do you have about the middle term of the product and the
product of the first and last terms of the binomial?
Observe the following examples:
a. (x – 3)2 = (x)2 – 3x – 3x + (3)2 c. (3x + 4y)2 =(3x)2 + 12xy + 12xy +
(4y)2 =
x2 – 2(3x) + 9 = 9x2 + 2(12xy) +
16y2
=
x2 – 6x + 9 = 9x2 + 24xy + 16y2
b. (x + 5)2 = (x)2 + 5x + 5x + (5)2
=
x2 + 2(5x) + 25
= x2 + 10x + 25
Don’t
o
F rget!
Remember:
•
Product rule •
Raising a power to a power
(am)(an) = am+n(am)n = amn
17
You can give this one as drill to the students.
1.(s + 4)2 = s2 + 8s + 16
6. (5d – 7d2t)2 = 25d2 – 70d3t + 49d4t2
2.(w – 5)2 = w2 – 10w + 25
7. (7q2w2 – 4w2)2 = 49q4w4 – 56q2w4 + 16w4
3.(e – 7)2 = e2 – 14e + 49
2 e – 6)2 = 4 e2 – 8e+36
3
9
9. ( 4 kj – 6)2 = 16 k2j2 – 48 kj + 36
5
25
5
4.(2q – 4)2 = 4q2 – 16q + 16
5.(3z + 2k)2 = 9z2 + 12zk + 4k2
The square of binomial, consists of:
a. the square of the first term;
b. twice the product of the first and last terms; and
c. the square of the last term.
8. (
10. [(x + 3) – 5]2 = x2 – 7x + 64
You can use problem no. 10 in the learning module to link the next topic
from the previous lesson to model squaring trinomials, allow them to do as a
group the paper cutting activity.
Provide hooking questions to the students before performing this activity.
Ask them how they will square a trinomial even without going to the process
of FOIL method. Activity 5 may be given as a group activity or by pair.
Remind the students that the folding they performed in creating vertical
creases should be equal to the folds that they will do to create horizontal
creases. After folding they can cut the creases to form different figures. They can use the measurement found in their learning modules.
Use process questions for the students to realize that the square of trinomial
can be modelled by a2 + b2 + c2 + 2ab + 2ac + 2bc. Provide more examples
to generate rules in squaring trinomials.
Remember that the square of a binomial is called a perfect square trinomial.
LET’S PRACTICE!
Square the following binomials using the pattern you have learned.
1.(s + 4)2 5.(3z + 2k)29. ( 4 kj – 6)2
5
2.(w – 5)2
6. (5d – 7d2t)2
10. [(x + 3) – 5]2
3.(e – 7)2
7. (7q2w2 – 4w2)2
( 2 e – 6)2
2
The square of a binomial is just one example of special products. Do the next activity to
discover another type of special product, that is squaring a trinomial.
4.(2q – 4)2
A ctivity 5
Description: Directions: Teacher’s Note and Reminders
Don’t
o
F rget!
8. DISCOVER ME AFTER!
(PAPER FOLDING AND CUTTING)
In this activity you will model and discover the pattern on how a trinomial is
squared that is (a + b + c)2. Investigate and observe the figure that will be
formed.
Get a 10” × 10” square paper. Fold the sides 7”, 3” and 1” vertically and make
crease. Using the same measures, fold horizontally and make creases. The
resulting figure should be the same as the figure below.
7
3
1
a
18
b
c
Teacher’s Note and Reminders
QU
?
NS
ES TI O
1.
2.
3.
4.
5.
6.
How many regions are formed? What are the dimensions of each
region in the figure?
What are the areas of each region?
If the side of the biggest square is replaced by a, how will you
represent its area?
If one of the dimensions of the biggest rectangle is replaced by b,
how will you represent its area?
If the side of the smaller square is replaced by c, how will you
represent its area?
What is the sum of the areas of all regions? Do you observe any
pattern in the sum of their areas?
Observe the following examples and take note of your observation.
a.(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
b.(m + n – d)2 = m2 + n2 + d2 + 2mn – 2md – 2nd
c.(3d + 2e + f)2 = 9d2 + 4e2 + f2 + 12de + 6df + 4ef
Don’t
Forget!
The square of a trinomial, consists of:
a. the sum of the squares of the first, second and last terms;
b. twice the product of the first and the second terms;
c. twice the product of the first and the last terms; and
d. twice the product of the second and the last terms.
LET’S PRACTICE!
1.(r – t + n)26. (15a – 4n – 6)2
Answer:
1. r2 + t2 + n2 – 2rt + 2rn – 2tn
2. e2 + 4a2 + q2 + 4ae + 2eq + 4q
3. m2 + a2 + y2 + 2ma – 2my – 2ay
4.4s2 + o2 + 16n2 + 4so – 16sn – 8on
Square the following trinomials using the pattern you have learned.
2.(e + 2a + q)27. (4a + 4b + 4c)2
6.
7.
8.
9.
225a2 + 16n2 + 36 – 120an +48n – 180a
16a2 + 16b2 + 16c2 + 24ab + 24ac + 24bc
81a4 +16b4 +9c4 +72a2b2 – 54a2c2- 24a2c2
2.25a4 + 5.29b2 + 1 – 6.9a2b + 3a2 – 4.6b
9 2 16 2
5.4i4 + 9a2 + 25n2 + 12i2a – 10i2n – 30an 10.
x +
y – 36 + 2xy + 9x – 16y
16
9
3.(m + a – y) (m + a – y)8. (9a2 + 4b2 – 3c2)2
4.(2s + o – 4n)29. (1.5a2 – 2.3b + 1)2
5.(2i2 + 3a – 5n)2
10. ( 3x + 4y -6)2
3
4
19
Emphasize to the students that the activity will help them model the product
of sum and difference of two terms. Make them realize that the area of the
resulting figure is the product of the sum and difference of two terms. The
resulting figure should be the same as the figure found below. A ctivity 6
TRANSFORMERS
Description: Directions: This activity will help us model the product of the sum and difference of
two terms (x – y) (x + y) and observe patterns to solve it easily.
Prepare a square of any measure; label its side as x. Cut a small square
of side y from any of its corner (as shown below). Answer the questions
that follow.
B
A
After doing the above activity, provide for more examples for the students to
see the pattern in finding the product of the sum and difference of binomials.
(Note: In the above activity, help the students realize that the dimension of
the rectangle is (x + y) by (x – y) and its area is x2 – y2.)
x
y
x
G
D
E
F
y
C
In terms of x and y, answer the following:
1. What is the area of the original big square (ABCD)?
2. What is the area of the small square (GFCE)?
3. How are you going to represent the area of the new figure?
x–y
Cut along the broken line as shown and rearrange the pieces to form a
rectangle.
1. What are the dimensions of the rectangle formed?
2. How will you get the area of the rectangle?
3. Represent the area of the rectangle that was formed. Do you see
any pattern in the product of the sum and difference of two terms?
Answers on lets practice.
1. w2 – 36
4. 16f2 – 9s2d2
7. L6o8v10 – 36e6
2. a2 – 16c2
5. 144x2 – 9
8.
3.16y2 – 25d2
6. 9s4r4 – 49a2
9. 4s2nq2m – 9d6k
Study the relationship that exists between the product of the sum and
difference of two terms and the factors and take note of the pattern
formed.
25 4 4
4 4
ga –
d
36
9
a.(x + y)(x – y) = x2 – y2
b.(a – b)(a + b) = a2 – b2
c.(m + 3)(m – 3) = m2 – 9
d.(w – 5)(w + 5) = w2 – 25
e.(2x – 5)(2x +5) = 4x2 – 25
10. (s + 2)2 – 16 = s2 + 4s – 12
The product of the sum and difference of two terms is the difference of the squares
of the terms. In symbols, (x + y)(x – y) = x2 – y2. Notice that the product is always a
binomial.
LET’S PRACTICE!
Multiply the following binomials using the patterns you have learned.
1.(w – 6)(w + 6)
3. (4y – 5d)(4y + 5d)
2.(a + 4c)(a – 4c)4.
(3sd + 4f)(4f – 3sd)
20
In activity no. 7, ask the students to do the solid figures the day before the
activity. Cubes must have the following sizes:
Solid figures:
1. One 3 in. x 3 in. cube using card board.
2. One 1 in. x 1 in. cube using card board.
3. Three prisms whose square base is 3 in. and height of 1 in.
4. Three prisms whose square base is 1 in. and height of 3 in.
Ask the students to calculate the volume of each solid figure. A. Step 1 (for prism)
d2)(
5
6
g2a2 +
6. (3s r + 7q)(3s r – 7q)
9. (2s q + 3d ) (2s q – 3d )
7. 2 2
n m
(l o v – 6e ) (l o v + 6e )
3 4 5
3
3 4 5
3k
n m
2
3
d2)
3k
10. [(s + 2)– 4][(s + 2) + 4]
3
The previous activity taught you on how to find the product of sum and difference
of two terms using patterns. Perform the next activity to discover another pattern in
simplifying expressions of polynomials.
A ctivity 7
CUBRA CUBE
Description: A cubra cube is a set of cubes and prisms connected by nylon. The task
is to form a bigger cube using all the figures provided. Your teacher will
help you how to form a cubra cube. After performing the activity, answer
the questions that follow.
a
b
2
3
8. (
2 2
a
B.
g2a2 –
5.(12x – 3)(12x + 3)
Note: The following are the patterns in creating the solid figures:
A. Step 1 (for cubes)
5
6
B.
a
a
b
Give more exercises to the students regarding the lesson, allow the students
to state in their own words the rule in cubing binomials based on the activity
and examples. After the discussion, have a short summary of all types of
special products the students have encountered.
b
b
QU
?
NS
ES TI O
1.
2.
3.
4.
5.
6.
7.
21
How many big cubes did you use? Small cubes?
How many different prisms do you have?
How many prisms are contained in the new cube?
What is the total volume of the new cube formed?
If the side of the big cube is marked as a and the smaller cube is
marked as b, what is the volume of each figure?
What will be the total volume of the new cube?
What are the dimensions of the new cube?
Let the students complete the Revised part of the IRF worksheet, but this
time they must have already realize and be able to correct the mistakes they
have on initial part.
To reinforce students understanding, let them do the web – based
exercises / games or you can have them the linking base game by group
(found below). The students will write the product of the branches where
the rectangle is attached. You can modify the example below to suit it your
learners.
To include valuing, relate the activity to an organization, by asking the
following questions:
1. What will happen to the web if one of your groupmate wrote the wrong
product in one box?
2. What will happen in an organization if one of the members failed to do
his job?
This time let us go back to the gallery walk activity and focus to case 3 which is an
example of a cube of binomial (x + y)3 or (x + y)(x + y)(x + y) and (x – y)3 or (x – y)(x – y)
(x – y).
To find the cube of a binomial of the form (x + y)3:
a. Find the cube of each term to get the first and the last terms.
(x)3, (y)3
b. The second term is three times the product of the square of the first term
and the second term. 3(x)2(y)
c. The third term is three times the product of the first term and the square of
the second term. 3(x)(y)2
Hence, (x + y)3 = x3 + 3x2y + 3xy2 + y3
To find the cube of a binomial of the form (x – y)3:
a. Find the cube of each term to get the first and the last terms.
(x)3, (-y)3
b. The second term is three times the product of the square of the first term
and the second term. 3(x)2(-y)
c. The third term is three times the product of the first term and the square of
the second term. 3(x)(-y)2
3x + 2
3x + 2
4x – 2y + 6
9x2 – 4
Hence, (x – y)3 = x3 – 3x2y + 3xy2 – y3
3x – 2
A ctivity 8
IRF WORKSHEET
3x – 2
Description: 4x – 2y + 6
3x – 2
Note: You can use the video lessons found in the learning modules for the
discussion of different types of special products.
Now that you have learned how to find the different special products,
using the “R” portion of the IRF Worksheet, answer the topical focus
question: What makes a product special? What patterns are involved
in multiplying algebraic expression?
Initial Answer
Revised Answer
Final Answer
Activity 9 (3 – 2- 1 chart) should be completed for you to know if there are
still some confusions about the lesson. This activity should be served as
bring home activity.
22
Teacher’s Note and Reminders
WEB – BASED ACTIVITY: DRAG AND DROP
Description: Now, that you have learned the various
special products, you will now do an
interactive activity which will allow you to
drag sets of factors and drop them beside
special products. The activity is available in
this website: http://www.media.pearson.com.
au/schools/cw/au_sch_bull_gm12_1/dnd/2_
spec.html.
QUESTIONS:
1. What special products did you use in the activity?
2. Name some techniques which you used to make the
work easier.
3. What generalizations can you draw out of the examples
shown?
4. Given the time constraint, how could you do the task
quickly and accurately?
A ctivity 9
3-2-1 CHART
Description: In this activity, you will be asked to complete the 3-2-1 Chart regarding
the special products that you have discovered.
3-2-1 Chart
Three things I found out:
1. _____________________________________________________
2. _____________________________________________________
3. _____________________________________________________
Two interesting things:
1. _____________________________________________________
2. _____________________________________________________
One question I still have:
1. _____________________________________________________
Don’t
Forget!
23
As concluding activity for process, ask the students to complete the chart
above by giving the different types of special products and state its step,
they can add box if necessary. This activity may be given as bring home
activity.
A ctivity 10
Description:
Teacher’s Note and Reminders
Directions: WHAT’S THE WAY, THAT’S THE WAY!
This activity will test if you really have understood our lesson by giving the
steps in simplifying expressions containing special products in your own
words.
Give the different types of special products and write the steps/process of
simplifying it. You may add boxes if necessary.
Don’t
o
F rget!
SPECIAL
PRODUCTS
SPECIAL
PRODUCTS
______________
______________
______________
______________
Now that you know the important ideas about how
patterns on special products were used to find the product of
an algebraic expressions, let’s go deeper by moving on to the
next section.
What to
to Understand
Understand
What
What to
to Understand
Understand
What
This part provides learners activities to further validate
and deepen their understanding on the applications of
special products.
Answers:
1.a.10 m by 6 m
b. (8 + 2)(8 – 2)m2 = (10)(6) = 60 m2, difference of two squares
c.(x + 2)(x – 2)m2
d. No, the area will be decreased by 4 m2
Now that you have already learned and identified the
different polynomials and their special products. You will now
take a closer look at some aspects of the topic and check if
you still have misconceptions about special products.
A ctivity 11
Directions:
2.a.64 in , 256 in
b. (74)(128) = 9472 ÷ 64 = 148 tiles
c.37
d. Either, because he will spend the same amount whichever tile he will
use.
2
2
DECISION, DECISION, DECISION!
Help each person decide what to do by applying your knowledge on
special products on each situation.
1.
Jem Boy wants to make his 8 meters square
pool into a rectangular one by increasing its
length by 2 m and decreasing its width by 2
m. Jem Boy asked your expertise to help him
decide on certain matters.
http://www.oyster.com/las-vegas/hotels/luxor-
hotel-and-casino/photos/square-pool-north-luxorhotel-casino-v169561/#
24
a.
b.
To check if the students still has difficulty in this topic, as bring home activity
ask the students to do the muddiest point (Activity 12). Process their
answers the next day for your assessment if they are ready for the next topic
c.
d.
Teacher’s Note and Reminders
A ctivity 12
Description: Directions: 1.
What will be the new dimensions of Jem Boy’s pool?
What will be the new area of Jem Boy’s pool? What special
product will be use?
If the sides of the square pool is unknown, how will you
represent its area?
If Jem Boy does not want the area of his pool to decrease, will
he pursue his plan? Explain your answer.
Emmanuel wants to tile his rectangular floor. He has two kinds
of tiles to choose from, one of which is larger than the other.
Emmanuel hired your services to help him decide which tile to use.
a.
What area will be covered by the 8” x 8” tile? 16” x 16” tile?
b. If the rectangular floor has dimensions of 74” x 128”, how
many small square tiles are needed to cover it?
c. How many square big tiles are needed to cover the rectangular
floor?
d. If each small tile costs Php 15.00 and each big tiles costs Php
60.00, which tile should Emmanuel use to economize in tiling
his floor? Explain why.
AM I STILL IN DISTRESS?
This activity will help you reflect about things that may still confuse you in
this lesson.
Complete the phrase below and write it on your journal.
The part of the lesson that I still find confusing is __________________
because _________________________________________________.
Don’t
o
F rget!
Let us see if by your problem will be solved doing the next activity, where the
muddiest point will be clarified.
25
Note: Before activity 13 should be provided, prepare the students by letting
them watch a video on solving composite areas or giving them examples of
this type of problems.
In this activity, you should let the students realize that unknown quantities
can be represented by any variable.
A ctivity 13
Description: BEAUTY IN MY TILE!
See different tile patterns on the flooring of a building and calculate the area
of the region bounded by the broken lines, then answer the questions below.
1.
Answers:
a.
1. a. x2 sq. units, y2 sq. units, xy sq. units
b.(x2 + 2xy + y2) sq. units
c. Squaring binomial
b.
c.
d.
2.a.x2 sq. unit, y2 sq. units
b.9y2 sq. units
c.(x2 – 9y2)sq. units
http://www.apartmenttherapy.com/tile-vaultmidcentury-rec-room-39808
2.
a.
For the portfolio entry by the students, have the students form as group and
do Activity 14, this must be done outside the class. Tell the students that
they can sketch the figure if they have no devices to use.
b.
c.
Teacher’s Note and Reminders
QU
?
NS
ES TI O
A ctivity 14
1.
2.
3.
4.
What is the area represented by
the big square? small square?
rectangles?
What is the total area bounded by
the region?
What special product is present in
this tile design?
Why do you think the designer of
this room designed it as such?
What is the area represented by
the big square? Small square?
What is the sum of all areas of
small squares?
If the small squares were to be
removed, how are you going to
represent the area that will be left?
What difficulties did you experience in doing the activity?
How did you use special products in this activity?
What new insights did you gain?
How can unknown quantities in geometric problems be solved?
WHERE IS THE PATTERN?
Descriptions: Take a picture/sketch of a figure that makes use of special products. Paste
it in a piece of paper.
Don’t
o
F rget!
QU
26
?
NS
ES TI O
1.
2.
3.
Did you find difficulty in looking for patterns where the concept of
special products were applied?
What special products were applied in your illustration?
What realization do you have in this activity?
As part of the concluding activity for process, have the student’s debate on
the answers of the questions found in Activity 15. You can have the class
form the different rules for debate.
As culminating activity of the students in this section, ask them to fill – up
the final part of the IRF worksheet, this may be assigned as bring home
activity. Tell them to compare their answers with the other and ask them
their realization in this topic.
A ctivity 15
Description: LET’S DEBATE!
Form a team of 4 members with your classmates and debate on the two
questions below. The team that can convince the other wins the game.
•
“Which is better to use in finding products, patterns or long
multiplication?”
•
“Which will give us more benefit in life, taking the short – cuts or
going the long way?
Teacher’s Note and Reminders
A ctivity 16
Description: Don’t
Forget!
IRF WORKSHEET
Now that you have learned the different special products, using the “F”
portion of the IRF Worksheet, answer the topical focus question: What
makes a product special? What patterns are involved in multiplying
algebraic expressions?
IRF Worksheet
Initial Answer
Revised Answer
Final Answer
Now that you have a deeper understanding of the topic, you are ready to do the tasks
in the next section.
What to
to Transfer
Transfer
What
What to
to Transfer
Transfer
What
Provide learners opportunities to apply their understanding on special
products activities that reflect meaningful and relevant problems/situations. Students will have the opportunity to see the relevance of their discussions
and its possible application in real – life.
Let us now apply your learning to real – life situations. You will be given a
practical task which will demonstrate your understanding.
27
Before doing the activity, ask the students’ to do a research on the uses and
importance of genetics in the study of human life. And give the following
definition and small discussions on genetics especially the heterozygous
and homozygous traits.
A ctivity 17
Description: MAKE A WISH
Concept of squaring binomials is used in the field of Genetics through
PUNNET squares. PUNNETT SQUARES are used in genetics to model
the possible combinations of parents’ genes in offspring. In this activity
you will discover how it will be used.
Investigate how squaring trinomials are applied in PUNNET squares and
answer the following questions.
• Genetics is the area of biological study concerned with heredity and with
Direction: the variations between organisms that result from it.
• Homozygous refers to having identical alleles (group of genes) for a single
trait. (SS)
• Heterozygous refers to having two different alleles (group of genes) for a One cat carries heterozygous, long-haired
traits (Ss), and its mate carries heterozygous,
single trait. (Ss)
Note: Capital letter denotes dominant traits, while small letter denotes
recessive traits. Dominant traits will come out in heterozygous.
Teacher’s Note and Reminders
QU
?
NS
ES TI O
S
s
S
SS
Ss
long-haired traits (Ss). To determine the
chances of one of their offspring having short
hair we can use PUNNET Squares.
s
Ss
ss
1.
2.
3.
4.
5.
6.
7.
Don’t
Forget!
What are the chances that the offspring is a long – haired cat? A
short – haired cat?
What are the different possible offsprings of the mates?
How many homozygous offsprings will they have? Heterozygous?
How is the concept of squaring binomials used in this process?
Do you think it is possible to use the process of squaring trinomials
in the field of genetics?
Create another model of PUNNET square using a human genetic
component. Explain the possible distribution of offsprings and
how squaring trinomials help you in looking for its solution.
Create your own PUNNET square using the concept of squaring
trinomials, using your dream genes.
Now that you have seen the different
patterns that can be used in simplifying polynomial
expressions, you are now ready to move to
the next lesson which is factoring. Observe the
different patterns in factoring that are related to
special products so that you can do your final
project, the making of packaging box.
28
Punnett square is named after
Reginald C. Punnett, who
devised the approach, and is used
by biologists to determine the
chances of an offspring’s having a
particular genotype. The Punnett
square is a tabular summary of
every possible combination of one
maternal allele with one paternal
allele for each gene being studied
in the cross.
Lesson 2:
Factoring
Lesson
What to
to Know
Know
What
Initially, begin with some interesting and challenging activities that
will enable the students to see the association of products to factors
and activate their prior knowledge on factoring.
2
2
Factoring
What to
toKnow
Know
What
Your goal in this section is to see the association of products to factors by doing the activities that
follows.
Before you start doing the activities in this lesson, first do this challenge.
As a review on basic concepts of factoring, allow the students to give
the different dimensions of rectangle they can create out of a square whose
area is 36 units squared (e.g. 18 and 2, 9 and 4), with this they will realize The figure below is a square made up of 36 tiles. Rearrange the tiles to create a
that the different factors of 36 are the dimensions of rectangle. Ask the topical rectangle, having the same area as the original square. How many such rectangles can you
create? What are your considerations in looking for the other dimensions? What mathematical
question to the students and the essential question.
“What expressions can be factored? How are patterns used in finding
the factors of an expression? How can unknown quantities in geometric
problems be solved?
To start the lesson perform Activity 1 by distributing thumbs up icon to the
students and allow them to paste it under the response column. Thumbs
up means a student has little mastery on the skills described and a thumbs
down signifies that the student has already mastered the skills. This will
serve as your guidance into the skills students still needed in this lesson.
You can add another row for skills if necessary
Teacher’s Note and Reminders
concepts did you consider in forming different dimensions? Why? Suppose the length of
one side is increased by unknown quantities (e.g. x) how could you possibly represent the
dimensions?
This module will help us break an expression into different factors and answer the
topical questions, “What algebraic expressions can be factored? How are patterns
used in finding the factors of algebraic expression? How can unknown quantities in
geometric problems be solved?
To start with this lesson, perform the activities that follow.
A ctivity 1
Don’t
o
F rget!
LIKE! UNLIKE!
Description: Directions: 29
This activity will help your teacher gauge how ready you are for this lesson
through your responses.
Answer all the questions below honestly by pasting the like or unlike thumb
that your teacher will provide you. Like means that you are the one being
referred and unlike thumb means that you have no or little idea about what
is being asked.
For conceptual map, ask the students to complete the I part of the IRF
sheet. Ask the group to keep their answer so that they can revisit it after
discussions.
SKILLS ACQUIRED
1. Can factor numerical expressions easily
2. Can divide polynomials
3. Can apply the quotient rule of exponents
4. Can add and subtract polynomials
5. Can work with special products
6. Can multiply polynomials
To activate prior knowledge of the students on the skills they will use in
this lesson perform Activity 3, at this point students should realize the
association of factors and products, and observe the different pattern that
will exist. Use the questions found after the activity..
RESPONSES
Before you proceed to the next topic, answer first the IRF form to
determine how much you know in this topic and see your progress.
Teacher’s Note and Reminders
A ctivity 2
IRF WORKSHEETS
Description: Complete the table by filling first the initial column of the chart. This
activity will determine how much you know about this topic and your
progress.
Initial
Express the following as
product of factors.
1. 4x2 – 12x =
_________________
2. 9m2 – 16n2 =
_________________
3. 4a2 + 12a + 9 =
_________________
4. 2x2 + 9x – 5 =
_________________
5. 27x3 – 8y3 =
_________________
6. a3 + 125b3 =
_________________
7. xm + hm – xn – hn =
_________________
Don’t
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30
Revise
Final
Answers Key
MESSAGE FROM THE KING
Activity 3
“FACTORING IS THE REVERSE OF MULTIPLICATION”
Teacher’s Note and Reminders
A ctivity 3 (Product – Factor Association)
Description: This activity will give you an idea on how factors is associated with
products. You will match the factors in column A with the products in
column B and decode the secret message.
COLUMN A COLUMN B
1.4x (3x – 5)A. 6x2y2 + 3xy3 – 3xy2
2.3xy2(2x + y – 1)
F. x3 – 27
3.(x + y)(x – y)G. 4x2 – 9
4.(2x + 3)(2x – 3)
R. 4x2 + 12x + 9
5.(x – 5y)(x + 5y)U. 12x2 – 20x
6.(x + y)2E. 6x2 + x – 2
7.(2x + 3)2T. ac – ad + bc – bd
8.(x – 5y)2S. mr – nr + ms – ns
9.(x + 4)(x – 3)
C. x2 – y2
10.(2x – 1)(3x +2)I. 2x2 – x – 10
11.(x + 2)(2x – 5)
O. x2 – 10xy + 25y2
12.(x – 3)(x2 + 3x + 9)
N. x2 + x – 12
13.(x + 3)(x2 – 3x + 9)
H. x3 + 27
14.(a + b)(c – d)M. x2 + 2xy + y2
15.(m – n)(r + s)L. x2 – 25y2
16.(3x + 4)(3x – 4)
P. 9x2 – 16
17.(3x – 4)2V. 9x2 – 24x + 16
12 2 3 14 8 7 11 9 4
11 15
7 10 17 10 7 15 10
14 13 10
8 12
6 1 5 14 11 16 5 11 3 2 14 11 8 9
QU
ES TIO
31
?
NS
Don’t
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F rget!
1.
2.
3.
4.
What are your observations on the expression in column A?
Compare them with those in column B.
Do you see any pattern?
Are the two expressions related?
Why is it important to know the reverse process of multiplication?
What to
to Process
Process
What
What did you discover between the relationship of products and its factors?
You have just tried finding out the relationship between factors and their product. You
can use this idea to do the next activities.
These are the enabling activities / experience that the learner will have to go
through to validate their observations in the previous activity. Interactive activities
are provided for the students to check their understanding on the lesson.
What you will learn in the next session will also enable you to do the final
project which involves model and lay – out making of a packaging box.
Start the discussion by defining factoring, you can use the questions found
in learning modules.
Allow the students to perform Activity 4, and use this as springboard in the
discussion of factoring by greatest common monomial factor. The students
will present their outputs in front. Use the questions to process the answer
of the students. Tell the students when to use and not to use this type of
factoring. Emphasize that this type of factoring should be use first before
applying any type of factoring.
Give examples of this type factoring after the activity. You can use the
examples found in learning module.
What to
to Process
Process
What
The activity that you did in the previous section will help you understand the
different lessons and activities you will encounter here.
The process of finding the factors of an expression is called factoring, which
is the reverse process of multiplication. A prime number is a number greater than
1 which has only two positive factors: 1 and itself. Can you give examples of prime
numbers? Is it possible to have a prime that is a polynomial? If so, give examples.
The first type of factoring that you will encounter is Factoring the Greatest
Common Monomial Factor. To understand this let us do some picture analysis.
Teacher’s Note and Reminders
A ctivity 4
FINDING COMMON
Description: Your task in this activity is to identify common things that are present in
the three pictures.
Don’t
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F rget!
http://k-pop-love.tumblr.com/post/31067024715/
eating-sushi
http://blog.ningin.com/2011/09/04/10-idols-and-groups-pigging-out/
QU
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1.
2.
3.
4.
32
What are the things common to these pictures?
Are there things that make them different?
Can you spot things that are found on one picture but not on the
other two?
What are the things common to two pictures but not on the
other?
Note to the teacher: Emphasize that the greatest common monomial factor
should be divided to all terms of the expression and not only to its first term
and that the number of terms of the other factor is equal to the number of
terms the polynomial contains.
The above activity gave us the idea about the Greatest Common Monomial Factor
that appears in every term of the polynomial. Study the illustrative examples on how
factoring the Greatest Common Monomial Factor is being done.
Teacher’s Note and Reminders
Factor 12x3y5 – 20x5y2z
a. Find the greatest common factor of the numerical coefficients.
The GCF of 12 and 20 is 4.
b.
Find the variable with the least exponent that appears in each term of the
polynomial.
x and y are both common to all terms and 3 is the smallest exponent for x
and 2 is the smallest exponent of y, thus, x3y2 is the GCF of the variables.
The product of the greatest common factor in (a) and (b) is the GCF of the
polynomial.
Hence, 4x3y2 is the GCF of 12x3y5 – 20x5y2z.
c.
Don’t
Forget!
d.
To completely factor the given polynomial, divide the polynomial by its GCF, the
resulting quotient is the other factor.
Thus, the factored form of 12x3y5 – 20x5y2z is 4x3y2(3y3 – 5x2z)
Below are other examples of Factoring the Greatest Monomial Factor.
a.8x2 + 16x  8x is the greatest monomial factor. Divide the polynomial by 8x
to get the other factor.
8x(x + 2) is the factored form of 8x2 + 16x.
b.12x5y4 – 16x3y4 + 28x6  4x3 is the greatest monomial factor. Divide the given
expression by the greatest monomial factor to get the other factor.
Polynomial
6m + 8
Greatest Common
Monomial Factor
(CMF)
Quotient of
Polynomial and
CMF
Factored
Form
2
3m + 4
2 (3m + 4)
4mo (3m + o)
12m o + 4mo
4mo
3m + o
27d4o5t3a6 – 18d2o3t6 – 15d6o4
3d2o3
9d2o2t3a6 – 6t6 – 5d4
4(12) + 4(8)
4
(12 + 8)
12WI3N5 – 16WIN + 20WINNER
4WIN
2 2
2
2
Thus, 4x3 (3x2y4 – 4y4 + 7x3) is the factored form of the given expression.
Complete the table to practice this type of factoring.
Polynomial
2
6m + 8
4(12 + 8)
Greatest Common
Monomial Factor
(CMF)
Quotient of
Polynomial and
CMF
2
3m + 4
4mo2
4(12) + 4(8)
12WI3N5 – 16WIN + 20WINNER
33
4
2 (3m + 4)
4mo2 (3m + o)
9d2o2t3a6 – 6t6 – 5d4
27d4o5t3a6 – 18d2o3t6 – 15d6o4
Factored
Form
Before doing the activity for factoring difference of two squares, ask the
students why the difference of two squares was given such name.
To start the discussion you can use number pattern to see the relationship
of factors to product. You may bring back the students to multiplying sum
and difference of binomials in special product to see how factors may be
obtained. Students should realize that factors of difference of two squares
are sum and difference of binomials.
Ask students to generate rule in factoring difference of two squares.
For paper cutting, students must realize that the area of the new figure
formed is the difference of the area of the two squares, which is (a2 – b2) and
that the dimensions of the rectangle formed are (a + b) x (a – b).
This activity may be done by pair or as a group.
Now that you have learned how to factor polynomials using their greatest common
factor we can move to the next type of factoring, which is the difference of two squares.
Why do you think it was given such name? To model it, let’s try doing the activity
that follows.
A ctivity 5
INVESTIGATION IN THE CLASSROOM
Description: This activity will help you understand the concepts of difference of two
squares and how this pattern is used to solve numerical expressions.
Investigate the number pattern by comparing the products then write your
generalizations afterwards.
NUMBER PATTERN:
a. (11)(9) = (10 + 1)(10 – 1) = 100 – 1 =
b. (5)(3) = (4 + 1)(4 – 1) = 16 – 1 =
c. (101)(99) = (100 + 1)(100 – 1) = 10000 – 1 =
d. (95)(85) = (90 + 5)(90 – 5) = 8100 – 25 =
e.(n – 5)(n + 5) =
Teacher’s Note and Reminders
How do you think products are obtained? What are the different techniques used to
solve for the products?
What is the relationship of the product to its factor? Have you seen any pattern in this
activity?
For you to have a clearer view of this type of factoring, let us have paper folding
activity again.
A ctivity 6
Don’t
o
F rget!
INVESTIGATION IN PAPER FOLDING
Description: Directions:
This activity will help you visualize the pattern of difference of two
squares.
A
1.
2.
3.
4.
34
Get a square paper and label the sides as a.
Cut – out a small square in any of its corner
and label the side of the small square as b.
Cut the remaining figure in half.
Form a rectangle
B
G
C
E
F
D
ES TI O QU
?
NS
You can use the examples found in learning module for the discussion. Give
more examples if necessary.
(Note: Remind students to use first factoring greatest common monomial
factor if applicable before factoring it through difference of two squares)
Teacher’s Note and Reminders
1.
2.
3.
4.
5.
What is the area of square ABDC?
What is the area of the cut – out square GFDE?
What is the area of the new figure formed?
What is the dimension of the new figure formed?
What pattern can you create in the given activity?
For you to have a better understanding about this lesson, observe how the
expressions below are factored and observe the relationships of the term with each other.
a. x2 – y2 = (x + y)(x – y)d.
16a6 – 25b2 = (4a3 – 5b)(4a3 + 5b)
b.4x2 – 36 = (2x + 6)(2x – 6)
c.a2b4 – 81 = (ab2 – 9)(ab2 + 9)
QU
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1.
2.
3.
4.
5.
6.
7.
e. ( 9 r4 – 1 t2 n6 ) = ( 3 r2 + 1 tn3)( 3 r2 – 1 tn3)
16
25
4
5
4
5
What is the first term of each polynomial?
What is the last term of each polynomial?
What is the middle sign of the polynomial?
How was the polynomial factored?
What pattern is seen in the factors of the difference of two terms?
Can all expressions be factored using difference of two squares?
Why or why not?
When can you factor expressions using difference of two squares?
Remember the factored form of a polynomial that is a difference of two
squares is the sum and difference of the square roots of the first and last terms.
Don’t
o
F rget!
•
35
4x2 – 36y2  the square root of 4x2 is 2x and the square root of 36y2 is 6y. To
write their factors write the product of the sum and difference of the square
roots of 4x2 – 36y2, that is (2x + 6y)(2x – 6y).
To check students understanding on factoring difference of two squares, ask
them to make pairs of square terms and factor it after. Students can give as
many pairs of difference of two square as they can create.
(Note: Teachers must see to it that students must form difference of two
squares)
Example Answer: 81m4 – 121c4 = (9m2 – 11c2) (9m2 + 11c2)
A ctivity 7
PAIR MO KO NYAN!
Description: This game will help you develop your factoring skills by formulating your
problem based on the given expressions. You can integrate other factoring
techniques in creating expressions. Create as many factors as you can.
Directions: Form difference of two squares problems by pairing two squared quantities
then find their factors. (Hint: You can create expressions that may require
the use of the greatest common monomial factor)
To start with factoring sum or difference of two cubes, allow students to
multiply (a + b)(a2 + ab + b2) and (a – b)(a2 + ab + b2). They should get (a3 + b3)
and (a3 – b3) respectively as the product. Ask the process question to the
students and help them see the pattern in factoring sum or difference of two
cubes. Guide them to generate the rule in factoring sum or difference of two
cubes.
Teacher’s Note and Reminders
You have learned from the previous activity how factoring the difference of two
squares is done and what expression is considered as the difference of two squares. We
are now ready to find the factors of the sum or difference of two cubes. To answer this
question, find the indicated product and observe what pattern is evident.
a.(a + b)(a2 – ab + b2)
b.(a – b)(a2 + ab + b2)
Don’t
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F rget!
What are the resulting products? How are the terms of the products related to the terms
of the factors? What if the process was reversed and you were asked to find the factors of the
products? How are you going to get the factor? Do you see any common pattern?
36
Use Activity 8 (Road Map to Factor) as guide in factoring sum or cubes of
binomials, this will give the students steps in factoring such expression. Give
more examples of sum or difference of two cubes and factor it to firm – up the
understanding of the students in factoring this expression.
Note: Remind the students to use first factoring by greatest common
monomial factor before applying this type of factoring if necessary.
A ctivity 8
ROAD MAP TO FACTOR
Answer the following problems by using the map as your guide.
Is the given
expression a sum
or difference of two
cubes?
Answers to problem:
1. (x3 – y3) unit cube =(x – y)(x2 + xy + y2)
2. (x3 + y3) unit cube = (x + y)(x2 – xy + y2)
If No
Use other factoring technique/method
If
Yes
Are the binomials sums or
differences of two cubes?
Teacher’s Note and Reminders
If DIFFERENCE
If
Sum
1. What are the cube roots of the first and
last terms?
2. Write their sum as the first factor. (x +
y).
3. For the second factor, get the trinomial
factor by:
a. Squaring the first term of the first
factor;
b. Subtracting the product of the first
and second terms of the first factor.
c. Squaring the last term of the first
factor
4. Write them in factored form.
(x + y)(x2 – xy + y2)
Don’t
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37
1. What are the cube roots of the first
and last terms?
2. Write their difference as the first
factor. (x – y).
3. For the second factor, get the
trinomial factor by:
a. Squaring the first term of the first
factor;
b. Adding the product of the first and
second terms of the first factor.
c. Squaring the last term of the first
factor
4. Write them in factored form.
(x – y)(x2 + xy + y2)
1.
Represent the volume of this figure. What
is the factored form of the volume of given
figure?
2.
What are the volumes of the cubes? If the
cubes are to be joined to create platform
for a statue, what will be the volume of
the platform? What are the factors of the
volume of the platform?
x
y
To start factoring perfect square trinomials, use algebra tiles to model it.
This activity will give the students picture of perfect square trinomials. See
to it that the students will produce a square.
Discuss the answers of the students on process questions. Point out
that the result of squaring binomial is a perfect square trinomial. At this point
students should see the pattern of factoring perfect square trinomials and be
able to generate the rule in factoring such polynomials.
Discuss when an expression is a perfect square. Do the perfect hunt
activity to check the students understanding in identifying perfect square
trinomials.
Teacher’s Note and Reminders
A ctivity 9
Let’s tile it up!
Directions: Prepare the following:
1.
2.
3.
1.
QU
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Form squares using:
•
•
•
•
•
1 big square tile, 2 rectangular tiles and 1 small square.
1 big square tile, 4 rectangular tiles and 4 small squares.
1 big square tile, 6 rectangular tiles and 9 small squares.
4 big square tiles, 4 rectangular tiles and 1 small square.
4 big square tiles, 8 rectangular tiles and 4 small squares.
1.
2.
3.
4.
How will you represent the total area of each figure?
Using the sides of the tiles, write all the dimensions of the squares.
What did you notice about the dimensions of the squares?
Did you find any pattern in their dimensions? If yes, what are
those?
How can unknown quantities in geometric problems be solved?
5.
4 big squares measuring 4” × 4” and represent each square as x2.
8 rectangular tiles with measures of 4” × 1” and represent it as x.
16 small squares whose measures is 1” × 1” and represent this as
The polynomials formed are called perfect square trinomials.
Perfect square trinomial is the result of squaring a binomial. A perfect square
trinomial has first and last terms which are perfect squares and a middle term which is
twice the product of the square root of first and last terms.
Don’t
o
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38
Answer to Activity 10
A ctivity 10
Description: PERFECT HUNT
Look for the different perfect square trinomials found in the box. Answers
might be in diagonal, horizontal or vertical in form.
10x
81
18x
x2
4
15x
16x2
-24x
9
10x
28x
4x2
-16x
16
15x
25
49x2
16x
49
8x
16
24x
9
25
14x
8x
40x
30x
10x
7x
x2
12x
25x2
40
12x2
2
To factor perfect square trinomials:
Teacher’s Note and Reminders
2
a. Get the square root of the first and last terms.
b. List down the square root as sum/difference of two terms as the case may be.
You can use the following relationships to factor perfect square trinomials:
(First term)2 + 2(First term)(Last term) + (Last term)2 = (First term + Last term)2
(First term)2 – 2(First term)(Last term) + (Last term)2 = (First term – Last term)2
Remember to factor out first the greatest common monomial factor before
factoring the perfect square trinomial.
Don’t
o
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39
Ex. 1. Factor n2 + 16n + 64
Solution:
a. Since n2 = (n)2 and 64 = (8)2, then both the first and last terms are
perfect squares. And 2(n)(8) = 16n, then the given expression is a
perfect square polynomial.
b. The square root of the first term is n and the square root of the last
term is 8, then the polynomial is factored as (n + 8)2.
After the above, the rule in factoring perfect square trinomial may be discuss
you can use the examples in learning module. Examples of factoring perfect
square trinomials should be given to ensure mastery.
Answers on exercise:
a.(m + 6)2 e. (7a – 6)2 i. 2(3h + 1)2
b.(4d – 3)2 f. (11c2 + 3)2 j. 5f2(2f – 3)2
c.(a2b – 3)2 g. (5r + 8)2
d.(3n + 5d)2 h. ( 1 z + 2 )2
3
4
Teacher’s Note and Reminders
Ex. 2. Factor 4r2 – 12r + 9 Solution:
a. Since 4r2 = (2r)2 and 9 = (3)2, and since – 12r = 2(2r)(3) then it
follows the given expression is a perfect square trinomial.
b. The square root of the first term is 2r and the square root of the last
term is 3 so that its factored form is (2r – 3)2.
Ex. 3. Factor 75t3 + 30t2 + 3t
Solution:
a. Observe that 3t is common to all terms, thus, factoring it out first we
have:
3t (25t2 + 10t + 1)
b. Notice that 25t2 = (5t)2 and 1 = (1)2, and 10t = 2(5t)(1), then 25t2 + 10t +
1 is a perfect square trinomial.
c. Factoring 25t2 + 10t + 1 is (5t + 1)2, thus, the factors of the given
expression are 3t (5t + 1)2.
Explain why in Example 3, (5t + 1)2 is not the only factor. What is the
effect of removing 3t?
Exercises
Supply the missing term to make a true statement.
m2 + 12m + 36 =
(m + ___ )2
b.16d – 24d + 9 =
( 4d – ___)2
c.a4b2 – 6abc + 9c2 =
(a2b ___ ___)2
d.9n2 + 30nd + 25d2
=
(___ __ 5d)2
e.49g2 – 84g + 36
=
(___ __ ___)2
f.121c4 + 66c2 + 9
=
(___ __ ___)2
g.25r2 + 40rn + 16n2
=
(___ __ ___)2
h.1 x2 + 1 x + 4 16
3
9
i.18h2 + 12h + 2
=
(__ __ __)2
=
2 (___ __ ___)2
j.20f 4 – 60f 3 + 45f 2
=
___ (2f __ ___)2
a.
2
Don’t
Forget!
Is q2 + q – 12 a perfect square trinomial? Why?
Are all trinomials perfect squares? How do we factor trinomials that are not perfect
squares?
In the next activity, you will see how trinomials that are not perfect squares are
factored.
40
Give examples of quadratic trinomials that are not perfect square. Ask
them to factor it. This will make the students realize that there are some
trinomials that are not factorable using perfect square trinomials. Use this
as springboard before proceeding to activity 11.
Note: Make sure to it that the students will form rectangle as their figure. Ask them to compare the dimensions of the figure formed in this activity and
activity 9.
Teacher’s Note and Reminders
A ctivity 11
TILE ONCE MORE!!
Description: You will arrange the tiles according to the instructions given to form a
polygon and find its dimensions afterwards.
Directions: 1. Cut – out 4 pieces of 3 in. by 3 in. card board and label each as x2
representing its area.
2. Cut – out 8 pieces of rectangular cardboard with dimensions of 3
in. by 1 in. and label each as x representing its area.
3. Cut – out another square measuring 1 in. by 1 in. and label each
as 1 to represent its area.
Form rectangles using the algebra tiles that you prepared. Use
only tiles that are required in each item below.
a.
b.
c.
d.
e.
QU
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ES TIO
1 big square tile, 5 rectangular tile and 6 small square tiles.
1 big square tile, 6 rectangular tiles and 8 small square tiles.
2 big square tiles, 7 rectangular tiles and 5 small square tiles.
3 big square tiles, 7 rectangular tiles and 4 small square tiles.
4 big square tiles, 7 rectangular tiles and 3 small square tiles.
What is the total area of each figure?
2. Using the sides of the tiles, write all the dimensions of the
rectangles.
3. How did you get the dimensions of the rectangles?
4. Did you find difficulty in getting the dimensions?
Based on the previous activity, how can the unknown quantities in geometric
problems be solved?
If you have noticed there are two trinomials that were formed in the preceding
activity, trinomials that contains numerical coefficient greater than 1 in its highest degree
and trinomials whose numerical coefficient in its highest degree is exactly 1.
Don’t
o
F rget!
Let us study first how factoring trinomials whose leading coefficient is 1 being
factored.
Ex. Factor p2 + 5p + 6
Solution: a. List all the possible factors of 6.
Factors of 6
2
41
3
6
1
-2
-3
-6
-1
Give examples of general trinomials whose leading coefficient is 1. You can
use trial and error in factoring these examples. Use the examples found in
learning module. Giving more examples is highly suggested. You can ask
the students to generalize how factoring of this trinomial is attained.
Remind them again that they should use factoring by greatest common
monomial factor using this type of factoring, if applicable.
Teacher’s Note and Reminders
b. Find factors of 6 whose sum is 5.
•
2+3=5
•
6+1=7
•
(-2) + (-3) = -5
•
(-6) + (-1) = -7
c. Thus, the factor of p2 + 5p + 6 = (p + 2)(p + 3) Ex.Factor v2 + 4v – 21
Solution: a. List all the factors of – 21
Factors of -21
-3
7
-7
3
-21
1
-1
21
b. Find factors of -21 whose sum is 4.
•
-3 + 7 = 4
•
-7 + 3 = -4
•
-21 + 1 = -20
•
-1 + 20 = 19
c. Hence, the factors of v2 + 4v – 21 = (v – 3)( + 7)
Factor 2q3 – 6q2 – 36q, since there is a common monomial factor, begin by
factoring out 2q first, rewriting it, you have 2q (q2 – 3q – 18).
a.
Don’t
o
F rget!
b.
c.
42
Listing all the factors of – 18.
Factors of -18
-1
-2
-3
-18
-9
-6
18
9
6
1
2
3
Since – 6 and 3 are the factors whose sum is – 18, then the binomial factors
of q2 – 3q – 18 are (q – 6)(q + 3).
Therefore, the factors of 2q3 – 6q – 36q are 2q(q – 6)(q + 3).
To check students understanding on this factoring technique, you can do
the bingo game. Write on a strip the polynomials below and place them on
container. Draw the strip and read it in class, give the students time to factor
the polynomials.
n2 – n – 20
n2 + 5n + 6
n2 – 4n – 32
n2 – n – 42
n2 + 9n + 18
n2 + 11n + 18
n2 + 17n + 72
8. n2 – 12n + 35
9. n2 – 8n – 48 10. n2 + 14n – 32
11. n2 – 17n + 72
12. n2 + 9n + 8
13. n2 + 10n + 24
14. n2 – 2n – 48
To factor trinomials with 1 as the numerical coefficient of the leading term:
a. factor the leading term of the trinomial and write these as the leading
term of the factors;
b. list down all the factors of the last term;
c. identify which factor pair sums up to the middle term; then
d. write factor pairs as the last term of the binomial factors.
15. n2 + 11n + 24
NOTE: ALWAYS PERFORM FACTORING USING GREATEST COMMON
MONOMIAL FACTOR FIRST BEFORE APPLYING ANY TYPE OF
FACTORING.
A ctivity 12
Teacher’s Note and Reminders
FACTOR BINGO GAME!
Description: Bingo game is an activity to practice your factoring skills with speed
and accuracy.
Instruction: On a clean sheet of paper, draw a 3 by 3 grid square and mark the
center as FACTOR. Pick 8 different factors from the table below and
write the in the grid. As your teacher reads the trinomial, you will locate
its factors and marked it x. The first one who makes the x pattern
wins. (n + 4)(n – 5)
(n + 2)(n + 9)
(n – 8)(n – 9)
(n + 2)(n + 3)
(n (n
+ +9)(n
8)
4)(n +
– 5)
(n (n
+ +1)(n
8)
2)(n +
+ 9)
(n (n
– –8)(n
4)
8)(n +
– 9)
(n – 7)(n – 5)
(n (n
+ –6)(n
4)
8)(n +
+ 4)
(n (n
– –7)(n
6)
7)(n +
– 5)
(n + 3)(n + 6)
(n – 2)(n + 16) (n + 3)(n + 8)
(n + 2)(n + 3)
(n + 3)(n + 6)
Don’t
o
F rget!
43
?
NS
ES TIO
QU
1.
2.
3.
4.
5.
6.
7.
Remember:
1.
2.
3.
4.
(n + 9)(n + 8)
(n + 1)(n + 8)
(n (n
– +12)(n
6)(n + +
4)4) (n
(n – 2)(n + 16)
(n + 3)(n + 8)
– 8)(n + 6)
How did you factor the trinomials?
What did you do to factor the trinomials easily?
Did you find any difficulty in factoring the trinomials? Why?
What are your difficulties? How will you address those
difficulties?
Give polynomials whose numerical coefficient of the leading term is not
1. Factor this using trial and error. Allow the students to stress out the
disadvantages that they have encountered in using this technique.
Introduce the factoring by grouping or the AC method after. Ask them to
compare the process.
Teacher’s Note and Reminders
What if the numerical coefficient of the leading term of the trinomial is not 1, can
you still factor it? Are trinomials of that form factorable? Why?
Trinomials on this form are written on the form ax2 + bx + c, where a and b are the
numerical coefficients of the variables and c as the constant term. There are many ways
of factoring these types of polynomials, one of which is by inspection.
Trial and error are being utilized in factoring this type of trinomials. Here is an
example:
Factors of:
Factor 6z2 – 5z – 6 through trial and error:
Give all the factors of 6z2 and – 6
6z2
(3z)(2z)
(6z)(z)
-6
(3)(-2)
(-3)(2)
(1)(-6)
( -1)(6)
Write the all possible factors using the values above and determine the middle
term by multiplying the factors.
Possible Factors
(3z – 2)(2z + 3)
(3z + 3)(2z – 2)
(3z – 3)(2z + 2)
(3z + 2)(2z – 3)
(3z + 1)(2z – 6)
(3z – 6)(2z + 1)
(6z + 3)(z – 2)
(6z – 2)(z +3)
(6z – 3)((z + 2)
(6z + 2)(z – 3)
(6z + 1)(z – 6)
(6z – 6)(z + 1)
Don’t
o
F rget!
Sum of the product of the outer
terms and the product of the inner
terms
9z – 4z = 5z
-6z + 6z = 0
6z – 6z = 0
-9z + 4z = -5z
-18z + 2z = -16z
3z – 12z = -9z
-18z + 3z = -15z
18z – 2z = 16z
12z – 3z = 9z
-18z + 2z = -16z
-36z + z = -35z
6z – 6z = 0
In the given factors, (3z + 2)(2z – 3) gives the sum of -5z, thus, making it as the
factors of the trinomial 6z2 – 5z – 36.
How was inspection used in factoring? What do you think is the disadvantage of
using this?
44
Give the example you have used above and solve it through factoring by
grouping. Provide for more examples.
Teacher’s Note and Reminders
Factoring through inspection is a tedious and a long process, thus, knowing another
way of factoring trinomial would be very beneficial in your study of this module.
Another way of factoring is through grouping or AC method. Closely look at the given
steps and compare it with trial and error.
Factor 6z2 – 5z – 6
1.
Find the product of the leading term and the last term.
6z2 – 5z – 6
(6z2)(-6) = -36z2
2. Find the factor of – 36z2 whose sum is – 5z.
-9z + 4z = -5z
3. Rewrite the trinomial as four – term expressions by replacing the middle
term by the sum factor.
6z2 – 9z + 4z – 6
4. Group terms with common factors.
(6z2 – 9z) + (4z – 6)
5. Factor the groups using greatest common monomial factor.
3z (2z – 3) + 2(2z – 3)
6. Factor out the common binomial and write the remaining factor as sum or
difference of binomial.
(2z – 3)(3z + 2)
Factor 2k2 – 11k + 12
1.
2.
3.
4.
5.
6.
Don’t
o
F rget!
45
Multiply the first and last terms.
(2k2)(12) = 24k2
Find the factors of 24k2 whose sum is -11k.
(-3k) + ( -8k) = -11k
Rewrite the trinomial as four – term expressions by replacing the middle
term by the sum factor.
2k2 – 3k – 8k + 12
Group terms with common factor
(2k2 – 8k) + (-3k + 12)
Factor the groups using greatest common monomial factor.
2k(k – 4) – 3(k – 4)
Factor out the common binomial and write the remaining factor as sum or
difference of binomial.
(k – 4)(2k – 3)
To practice the factoring skills of the students, do Activity 13 in class. Group
the students and distribute flaglet on each group. Ask one group to give a
factorable polynomial then let the other group factor it.
Extend the concept of factoring by grouping by applying it to polynomials
with four terms. You can use the examples on learning module. Perform
Activity 14 as a group after.
Teacher’s Note and Reminders
Factor 6h2 – h – 2
1.
2.
3.
Multiply the first and last terms.
(6h2)(-2) = -12h2
Find the factors of 12h2 whose sum is h.
(-4h) + ( 3h) = -h
Rewrite the trinomial as four – term expressions by replacing the middle
term by the sum factor.
6h2 – 4h – 3h – 2
4. Group terms with common factor
(6h2 – 3h) + (-4h – 2)
5. Factor the groups using greatest common monomial factor.
3h(2h – 1) – 2(2h – 1)
6. Factor out the common binomial, and write the remaining factor as sum
or difference of binomial.
(3h – 2)(2h – 1)
A ctivity 13
WE HAVE!
Description: Instruction: QU
?
NS
ES TI O
This game will help you practice your factoring skills through a game.
Form a group of 5. Your task as a group is to factor the trinomial that
the other group will give. Raise a flaglet if you have already factored the
trinomial and shout, We have it! The first group to get 10 correct answers
wins the game.
1.
2.
3.
Do you find difficulty in playing the game? Why?
What hindered you to factor the trinomial?
What plan do you have to address these difficulties?
Let’s extend!!
We can use factoring by grouping technique in finding the factors of a polynomial
with more than three terms.
Don’t
Forget!
Let’s try factoring 8mt – 12at – 10mh – 15ah
Solution: 1. Group terms with common factor.
(8mt – 12at) + ( -10mh – 15ah)
2. Factor out the greatest common monomial factor in each group.
4t(2m – 3a) – 5h(2m – 3a)  Why?
3. Factor out the common binomial and write the remaining factor as
sum or difference of binomial.
(2m – 3a)(4t – 5h)
46
Teacher’s Note and Reminders
Don’t
Forget!
Factor 18lv + 6le + 24ov + 8oe
Solution: 1. Group terms with common factor.
(18lv + 6le) + (24ov + 8oe)  Why?
2. Factor out the greatest common monomial factor in each group.
6l(3v + e) + 8o(3v + 3)  Why?
3. Factor out the common binomial and write the remaining factor as
sum or difference of two terms.
(3v + e)(6l + 8o)
A ctivity 14
Answers for Activity 14
1.4wt + 2wh + 6it + 3ih=
WITH
2.15te – 12he + 10ty – 8hy = THEY
3. hv + av + he + ae=
HAVE
4.10ti – 8ts – 15hi + 12hs =THIS
5.88fo + 16ro – 99fm – 18rm =FROM
6.7s + 35om + 9se + 45oe =SOME
7.42wa + 54wt + 56ha + 72ht=WHAT
8.36yu – 24ro + 12ou – 72yr =YOUR
9.72he + 16we + 27hn + 6wh =WHEN
10.26wr – 91or + 35od – 10wd=WORD
• Make a wrap – up of all the factoring that you discussed, ask the students
to differentiate it. You can use the graphic organizer for this activity. This
may serve as bring home activity.
• Peer mentoring maybe done to help students understand better the topic. Make sure to it that in every group there is one responsible student. You
may facilitate the mentoring or you can give the group free hand in doing
this activity. Have this activity
• To practice the factoring skills of the students with speed and accuracy
you can do flashcard drill by pair or as a team.
Description: Instruction: FAMOUS FOUR WORDS
This activity will reveal the most frequently used four - letter word (no
letter is repeated) according to world – English.org through the use of
factoring.
With your groupmates factor the following expressions by grouping and
write a four - letter word using the variable of the factors to reveal the 10
most frequently used four - letter word.
1.4wt + 2wh + 6it + 3ih
2.15te – 12he + 10ty – 8hy
3. hv + av + he + ae
4.10ti – 8ts – 15hi + 12hs
5.88fo + 16ro – 99fm – 18rm
6.7s + 35om + 9se + 45oe
7.42wa + 54wt + 56ha + 72ht
8.36yu – 24ro + 12ou – 72yr
9.72he + 16we + 27hn + 6wh
10.26wr – 91or + 35od – 10wd
A ctivity 15
Description: Direction: 47
TEACH ME HOW TO FACTOR
(GROUP DISCUSSION /PEER MENTORING)
This activity is intended for you to clear your queries about factoring with
the help of your group mates.
Together with your group mates, discuss your thoughts and queries
regarding factoring. Figure out the solution to each others’ questions, you
may ask other groups or your teacher for help.
QU
ES TI O
Teacher’s Note and Reminders
?
1.
2.
3.
4.
NS
Ask the students to go back to IRF sheet and answer the Revise part.
Discuss the answer of the students in class.
A ctivity 16
Description: Instruction: ?
WITH A BLINK!
This is a flash card drill activity to help you practice with speed and
accuracy your factoring technique.
As a group you will factor the expressions flashed by your teacher, each
correct answer is given a point. The group with the most number of points
wins the game.
1.
2.
3.
NS
QU
ES TI O
What different types of factoring have you encountered?
What are your difficulties in each factoring technique?
Why did you face such difficulties?
How are you going to address these difficulties?
What techniques have you used to answer the questions?
What things did you consider in factoring?
Did you find difficulty in the factoring the polynomials? Why?
Now that we have already discussed the different types of factoring, let us
summarize our learning by completing the graphic organizer below.
A ctivity 17
GRAPHIC ORGANIZER
Description: To summarize the things you have learned, as a group, complete the
chart below. You may add box if necessary.
FACTORING
TECHNIQUES
Don’t
Forget!
48
___________
___________
___________
___________
___________
___________
___________
example
example
example
example
example
example
example
Teacher’s Note and Reminders
A ctivity 18
IRF REVISIT
Revisit your IRF sheet and revise your answer by filing in column 2 under REVISE.
Initial
Revise
Final
Express the following as product
of factors.
1.4x2 – 12x = ______
2.9m2 – 16n2 = ______
3.4a2 + 12a + 9 = ______
4.2x2 + 9x – 5 = ______
5.27x3 – 8y3 = ______
6. a3 + 125b3 = ______
7. xm + hm – xn – hn = ______
Don’t
Forget!
Now that you know the important ideas about this topic, let’s go deeper by moving on
to the next section.
What to
to Understand
Understand
What
What to
to Understand
Understand
What
Your goal in this section is to take a closer look at some aspects of the topic
and to correct some misconceptions that might have developed.
The following activities will check your mastery in factoring polynomials.
This part provides learners activities to further validate and deepen
their understanding on the applications of factoring and to check their
knowledge against misconception.
A ctivity 19
Answers for Activity 19
1. No, the factors of x2 – 4x – 12 are (x – 6)(x + 2), while 12 – 4x – x2 has
(2 – x)(6 + x) as its factors.
2. Yes, 3x(x2 – 4), thus, 3x3 – 12x = 3x (x – 2)(x + 2).
3. Difference of two squares is only applied if the middle operation is
minus.
4.a.(x + 2)(x + 2) = x2 + 4x + 4
b.(0.4x – 3)(0.4x + 3) = (0.16x2 – 9)
c.4x2y5 – 12x3y6 + 2y2 = 2y2 (2x2y3 – 6x3y4 + 1)
d.3x2 - 27 = 3 (x2 – 9) = 3 (x + 3)(x – 3)
Description: SPOTTING ERRORS
This activity will check how well you can associate between product and
its factors.
Instructions: Do as directed.
1. Your classmate asserted that x2 – 4x – 12 and 12 – 4x – x2 has the same
factors. Is your classmate correct? Prove by showing your solution.
2. Can the difference of two squares be applicable to 3x3 – 12x? If yes,
how? If no, why?
3. Your classmate factored x2 + 36 using the difference of two squares, how
will you make him realize that his answer is not correct?
4. Make a generalization for the errors found in the following polynomials.
a. x2 + 4 = (x + 2)(x + 2)
b.1.6x2 – 9 = (0.4x – 3)(0.4x + 3)
49
After performing activity 19 allow the students to revisit IRF worksheets and
discuss their answers as a group. You can ask them their thoughts in this
lesson.
c.4x2y5 – 12x3y6 + 2y2 = 2y2 (2x2y3 – 6x3y4)
d.3x2 – 27 is not factorable or prime
Teacher’s Note and Reminders
5.
A ctivity 20
Are all polynomial expressions factorable? Cite examples to defend your
answer.
IRF REVISIT
Revisit your IRF sheet and revise your answer by filing in column 3 under FINAL column.
Initial
Revise
Final
Express the following as product
of factors.
1.4x2 – 12x = ___
2.9m2 – 16n2 = ___
3.4a2 + 12a + 9 = ___
4.2x2 + 9x – 5 = ___
5.27x3 – 8y3 = ___
6. a3 + 125b3 = ___
7. xm + hm – xn – hn = ___
QU
Don’t
o
F rget!
?
NS
ES TI O
A ctivity 21
Description: Instruction: 1.
2.
What have you observed from your answers in your initial column?
Is there a big difference?
What realization can you make with regard to the relationship of
special products and factors?
MATHEMAGIC! 2 = 1 POSSIBLE TO ME
This activity will enable you to apply factoring to prove if 2 = 1.
Prove that 2 = 1 by applying your knowledge of factoring. You will need the
guidance of your teacher in doing such.
If a = b, Is 2 = 1?
Challenge the students by doing Activity 21. Guide them in doing this activity
and help them realize that there is an error in this process. After the activity
make the students realize that 2 = 1 is not possible. Cite the mistake in the
activity given. As bring home activity, allow the students to complete their
journal to reflect their experiences in this module/lesson.
a. Were you able to prove that 2 = 1?
b. What different factoring techniques have you used to arrive at the
solution?
c. What error can you pinpoint to disprove that 2 = 1?
d. What was your realization in this activity?
50
A ctivity 22
Description: Instruction: JOURNAL WRITING
This activity will enable you to reflect about the topic and activities you
underwent.
Reflect on the activities you have done in this lesson by completing the
following statements. Write your answers on your journal notebook.
A.
A ctivity 23
Description: Directions: LET’S SCALE TO DRAW!
In this activity you will draw plane figures to help you do the final project
after this module.
Using the skills you have acquired in the previous activity, follow your
teacher’s instruction.
To prepare the students for their final project do scaffold 1 and 2. This must
be a guided activity. Allow them to answer the process questions after and
discuss it in class.
Teacher’s Note and Reminders
1. Drawing Plane Figures
a. a square with a side which measures 10 cm.
b. a rectangle with a length 3 cm more than its width.
c. draw any geometric figure and label its dimensions
algebraically.
A discussion on scale drawing will follow. After the discussion, the
teacher will demonstrate the steps on how to do the following:
a. A tree is five meters tall. Using a scale of 1m:2cm, draw the
tree on paper.
b. The school’s flag pole is 10 m high. Using a scale of 2.5m:1
dm, draw a smaller version of the tree. Give the height of your
drawing.
Don’t
Forget!
51
Reflect on your participation in doing all the activities in this lesson
and complete the following statements:
•
I learned that I...
•
I was surprised that I...
•
I noticed that I...
•
I discovered that I...
•
I was pleased that I...
2. 3. The teacher will demonstrate how a cube can be made using a
square paper. Follow what your teacher did.
Teacher’s Note and Reminders
A ctivity 24 Model Making
Description: Creation of a solid figure out of a given plane figure and expressing it in
terms of factors of a polynomial.
1.
QU
?
NS
ES TI O
a.
b.
c.
d.
Don’t
Forget!
e.
f.
g.
Create a solid figure from the rectangular figure that was provided
by following the steps given.
a. Cut – out a 2 in by 2 in squares in all edges of a 12 in by 6 in
rectangle.
b. Fold all the sides upward.
c. Paste/tape the edges of the new figure.
What is the area of the original rectangle if its side is x units?
If the sides of the small squares is y, what expression represents
its area?
How will you express the area of the new figure in terms of the
variables stated in letters a and b?
What is the dimension of the new figure formed? How about the
volume of the solid?
If the value of x = 4 cm and the value of y = 1 cm, what will be the
dimension of the new figure? Its area? Its volume?
How did factoring help you find the dimensions of the new figure
formed? the area? the volume?
What did you learn from this activity?
How can unknown quantities in geometric problems be solved?
What new realizations do you have about the topic? What new connections have you
made for yourself?
What to
to Transfer
Transfer
What
Perform activity 25 to prepare the students for the final project. For students culminating project ask them to do a packaging box and
they will assume the role of member of a design department of a packaging
company. Present to the students the rubrics on how they will be graded in
this project. Students will present their output in front of the class. You may
involve the class in evaluating their works. This culminating project must be
done by group. You can have a display of their outputs.
Now that you have a deeper understanding of the topic, you are ready to do the tasks
in the next section.
What to
to Transfer
Transfer
What
Your goal in this section is apply your learning to real life situations. You will be
given a practical task which will demonstrate your understanding in special products
and factoring.
52
SAMPLE RUBRIC FOR THE TRANSFER ACTIVITY
A ctivity 25 I BRING MY TRASH HOME
How did you find the performance task? How did the task help you see the real world application of
the topic?
Description: CRITERIA
Outstanding
4
Satisfactory
3
Developing
2
Beginning
1
Explanation of
the Proposal
(20%)
Explanations
and presentation
of the lay-out is
detailed and clear.
Explanations
and
presentation of
the lay-out is
clear.
Explanations
and
presentation of
the lay-out is
a little difficult
to understand
but includes
critical
components.
Explanations
and presentation
of the lay –out
is difficult to
understand
and is missing
several
components.
Accuracy of
Computations
(30%)
Utilization of
Resources
(20%)
The computations
done are accurate
and show
understanding of
the concepts of
special products
and factoring.
There is an
explanation for
every computation
made.
The
computations
done are
accurate
and show a
wise use of
the concepts
of special
products and
factoring.
The
computations
done are
erroneous
and show
some use of
the concepts
of special
products and
factoring.
The
computations
done are
erroneous and
do not show
wise use of the
concepts of
special products
and factoring.
Resources are
efficiently utilized
with less than
10% excess.
Resources are
fully utilized
with less than
10%-25%
excess.
Resources are
utilized but a
lot of excess.
Resources
are not utilized
properly.
The models
are wellcrafted and
useful for
understanding
the design
proposal. They
showcase
the desired
product.
The diagrams
and models
are less
useful in
understanding
the design
proposal
The diagrams
and models are
not useful in
understanding
the design
proposal.
The models
Appropriateness are well-crafted
of the Model
and useful for
(30%)
understanding the
design proposal.
They showcase
the desired
product and are
artistically done.
RATING
Perform the activity in preparation for your final output in this module.
In response to the school’s environmental advocacy, you are
required to make cylindrical containers for your trash. This is in support
of the “I BRING MY TRASH HOME!” project of our school. You will
present your output to your teacher and it will be graded according to the
following criteria: explanation of the proposal, accuracy of computations,
and utilization of the resources and appropriateness of the models.
A ctivity 26 PACKAGING ACTIVITY
Description: This activity will showcase your learning in this module. You will assume
the role of a member of a designing team that will present your proposal
to a packaging company.
The RER packaging company is in search for
the best packaging for a new dairy product that they
will introduce to the market. You are a member of the
design department of RER Packaging Company. Your
company is tapped to create the best packaging box that
will contain two identical cylindrical containers with the
box’s volume set at 100 in3. The box has an open top.
The cover will just be designed in reference to the box’s
dimensions. You are to present the design proposal for
the box and cylinder to the Chief Executive Officer of
the dairy company and head of the RER Packaging
department. The design proposal is evaluated according
to the following: explanation of the proposal, accuracy
of computations, utilization of the resources and
appropriateness of the models.
OVERALL
RATING
53
The first
commercial
paperboard
(not
corrugated)
box was
produced in
England in
1817
POST - TEST
1.
Which statement is true?
a. The square of a binomial is also a binomial.
b. The product of a sum and difference of two terms is a binomial.
c.
The product of a binomial and a trinomial is the square of a trinomial.
d. The terms of the cube of a binomial are all positive.
Answer: B
2.
Which of the following is NOT a difference of two squares?
1 4
a.
x –1
4
b.
x2 – 0.0001y4
c.1.6(x – 1)2 – 49
d.(x + 1)4 – 4x6
Answer: C, it is the only binomial that is not a difference of two squares
3.
Which of the following can be factored?
a.0.08x3 – 27y3
b.1.44(x2 + 1) – 0.09
c.24xy(x – y) + 5 (x + y)
d.0.027(x2 + 1)3 – 8
Answer: D, it is factorable by difference of two cubes
4. Which of the following values of k will make x2 – 5x + k factorable?
a.
5c.
-10
b.
12d.
-14
Answer: D
54
5.
If a square pool is to be made a rectangle such that the length is increased by 6 units and the width is decreased by 6 units,
what will happen to its area?
a. The area will increase by 12
b. The area will decrease by 12
c.
The area will increase by 36.
d. The area will decrease by 36
Answer: D
6. What is the surface area of the given cube below?
(x + 3y) cm
A.(6x + 18y) cm2
B.(x2 + 6xy + 9y2)cm2
C.(6x2 + 26xy + 54y2)cm2
D.(x3 + 9x2y + 27xy2 + 27y3)cm2
Answer: C
7.
Factor 16x4 – 625y16 completely.
a.(4x2 – 25y4)(4x2 + 25y4)
b.(4x4 – 25y8)(4x4 + 25y8)
c.(2x2 + 5y4)(2x2 – 5y4) (4x4 + 25y8)
d.(2x2 + 5y4)(2x2 – 5y4)(2x2 + 5y4)(2x2 + 5y4)
Answer: C, by factoring completely
8.
The area of a rectangular garden is (12x2 – 8x – 15)m2, what are its dimensions?
a.(3x – 5)m by (4x + 3)m
b.(6x + 5)m by (2x – 3)m
c.(6x – 3)m by (2x – 5)m
Answer: B
d.(12x – 15)m by (x + 1)
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9.
How much wood is needed in the window frame illustrated below?
m
e
a.(m + e)2 square units
b.(m2 + e2) square units
c.(m – e)(m – e) square units
d.(m + e)(m – e) square units
Answer: D
10.
If the area of a square garden is (4x2 – 12x + 9) square units, is it possible to solve its sides?
a. Yes, using factoring difference of two squares.
b. No, one of the sides must be given
c.
Yes, the area is a perfect square trinomial
d. No, the area is not factorable
Answer: C
11.
Which of the following is a possible base of a triangle whose area is
A.(2x – 9) meters
B.(4x – 6) meters
C.(2x + 1) meters
D.(4x – 3) meters
Answer: B
(2x2 – 6x + 9) square meters?
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12.
Liza factored the expression 15x2y3 + 10x4y + 5xy as 5xy (3xy2 + 2x3). Did Liza factor it correctly?
a. No, because 5xy is not the common factor.
b. Yes, because the last term is cancelled out
c.
Yes, there exist a common factor on all terms
d. No, because the last term when factored is 1 and should not be omitted
Answer: D
13.
Anne squared 3x + 4y as 9x2 + 16y2, which of the following statement is correct with the answer of your classmate?
a. The answer is correct, because to square a binomial distribute the exponent
b. The answer is wrong, because the product of squaring a binomial is a trinomial.
c.
The answer is correct, because the product of squaring a binomial is another binomial
d. The answer is wrong, because to square a binomial is to multiply the expression by 2.
Answer: B
14. Below is the solution of Rogelio in factoring 3x4 – 243:
3(x4 – 81)
(x2 – 9)(x2 + 9)
(x + 3)(x – 3)
Is the solution of Rogelio correct?
A. No, because the other factors was omitted.
B. No, because it lacks 3 as its factor.
C. Yes, because 3x4 – 243 is divisible by x + 3.
D. Yes, because the complete factorization of the expression is (x + 3)(x – 3)
Answer: A
USE THE DATA BELOW TO ANSWER THE QUESTIONS THAT FOLLOW.
15. A driver asked you to create a utility box with no top from a 12 in by 10 in piece of metal by cutting identical squares from each
corner and turning up the sides. The box must have a capacity of 96 in3.
If you are the driver, what are the standards you will look into the box?
I.
Appropriateness of the dimension
II. Artistic
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III. Durability
IV. Innovations
a. I & II
b. III & IV
Answer: C
c. I & III
d. II & IV
16.
Which of the following is the appropriate thing to do to ensure that the correct dimension of the box will be obtained?
a. Find a model for the box and measure it.
b. Measure the sides of the squares thoroughly to create a box.
c.
Make a trial and error until the desired capacity is obtained.
d. Find the dimension of the square to be cut through factoring and scale drawing.
Answer: D
17. Marie Fe ask your advice on what to do so that her heterozygous blue eyed dog will have a big chance of having a blue eyed
offspring, what advice could you give?
a. Bring her dog to an ob – gyne
b. Pair it with another blue eyed dog
c.
Pair it with a homozygous blue eyed dog
d. Pair it with a heterozygous blue eyed dog.
Answer: C, using the concept of Punnet square, a homozygous creature paired with another homozygous will have
a high chance of resulting into the desired genes.
18. As finance officer of RTN plantation, you were asked by the company to prepare a budget to fence 120 hectares of your
company’s lot. What will you do to minimize the use of fencing materials, knowing that the length is 1 hectare less than twice
the width?
a. Estimate the dimension of the lot.
b. Measure the dimension of the lot manually.
c.
Solve the dimension of the lot.
d. Hire an engineer to survey the lot.
Answer: C
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19. Your friend an event organizer approach you to seek for your help to arrange 80 chairs in a weeding and suit it in the demand
of the couple that the number of chairs in each rows is two less than the number of rows. How will you help your friend as to
not to consume too much time in arranging?
a. Make a trial and error of arrangement.
b. Make a plan of arrangement of the chairs.
c.
Ask chair renting company to resolve the problem
d. Used the data given and make an appropriate plan.
Answer: D
20. As the principal of a school, you asked an architect to prepare a blue print for new classroom that you plan to build. The
square classroom should have different areas for utilities (lavatory, CR, storage room and locker). What criteria will you use
to approve the blue print?
I.
Maximizing the area
II. Appropriateness of the location utilities.
III. Dimensions of classroom utilities.
IV. Uniqueness of design
a. I, II & III
b. I, II & IV
Answer: A
c.
d.
I, III & IV
II, III & IV
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SUMMARY
After completion of this module the students should have learned that products of some polynomials are obtained using the
different patterns, and these products are called special products. They must also learn the different examples of special products,
such as, the square of binomials, sum and difference of two terms, squaring trinomials, and cubing a binomial.
Students must have also realized that factor of different products can be obtained through the use of different patterns and
rules. They should already learned the different types of factoring such as: (1) Factoring by greatest common monomial factor, (2)
Factoring difference of two squares, (3) Factoring perfect square trinomials, (4) Factoring general trinomials, (5) Factoring the sum
or difference of two cubes, and (6) Factoring by grouping.
And at this point student must already understand and used the concepts of special products and factoring in the context of
real – life situations.
GLOSSARY OF TERMS USED IN THIS LESSON:
AREA – the amount of surface contained by a figure
COMPOSITE FIGURE – a figure that is made from two or more geometric figures
FACTOR – an exact divisor of a number.
GENETICS – is the area of biological study concerned with heredity and with the variations between organisms that result from it.
GEOMETRY – the branch of mathematics that deals with the nature of space and the size, shape, and other properties of figures
as well as the transformations that preserve these properties.
GREATEST COMMON MONOMIAL FACTOR – is the greatest factor contained in every term of an algebraic expression.
HETEROZYGOUS – refers to having two different alleles (group of genes) for a single trait.
HOMOZYGOUS – refers to having identical alleles (group of genes) for a single trait.
PATTERN – constitutes a set of numbers or objects in which all the members are related with each other by a specific rule.
PERFECT SQUARE TRINOMIAL – result of squaring a binomial.
PERIMETER – the distance around a polygon.
POLYNOMIAL – is a finite sum of terms each of which is a real number or the product of a numerical factor and one or more variable
factor raised to a whole – number powers.
PRODUCT – the answer of multiplication
PUNNET SQUARE - is a diagram that is used to predict an outcome of a particular cross or breeding experiment. And is used by
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biologist to determine the chance of an offspring's having a particular genotype.
SCALE DRAWING – a reduced or enlarged drawing whose shape is the same as an actual object that it represents.
VOLUME – the measure of space occupied by a solid body
REFERENCES AND WEBSITE LINKS USED IN THIS LESSON:
Oronce, O & Mendoza, M. (2003). Exploring Mathematics. Rex Book Store. Manila, Philippines.
Oronce, O & Mendoza, M. (2007). E – Math: Worktext in Mathematics First Year High School. Rex Book Store. Manila, Philippines.
Gamboa, Job D. (2010). Elementary Algebra. United Eferza Academic Publications. Bagong Lipa, Batangas City.
Ho, Ju Se T., et al. 21st Century Mathematics: First Year (1996). Quezon City: Phoenix Publishing House, Inc.,
2010 Secondary Education Curriculum: Teaching Guide for Mathematics II. Bureau of Secondary Education. Deped Central Office
http://en.wikipedia.org/wiki/Punnett_square
http://www.youtube.com/watch?v=u5LaVILWzx8
http://www.khanacademy.org/math/algebra/polynomials/e/factoring_polynomials_1
http://www.xpmath.com/forums/arcade.php?do=play&gameid=93
http://www.quia.com/rr/36611.html
http://www.coolmath.com/algebra/algebra-practice-polynomials.html
http://www.onlinemathlearning.com/algebra-factoring-2.html
http://www.youtube.com/watch?v=3RJlPvX-3vg
http://www.youtube.com/watch?v=8c7B-UaKl0U
http://www.youtube.com/watch?v=-hiGJwMNNsM
www.world–english.org
http://www.smashingmagazine.com/2009/12/10/how-to-explain-to-clients-that-they-are-wrong/
http://www.mathman.biz/html/sheripyrtocb.html
http://blog.ningin.com/2011/09/04/10-idols-and-groups-pigging-out/
http://k-pop-love.tumblr.com/post/31067024715/eating-sushi
http://www.apartmenttherapy.com/tile-vault-midcentury-rec-room-39808
http://onehouseonecouple.blogzam.com/2012/03/ master-shower-tile-progress/
http://www.oyster.com/las-vegas/hotels/luxor-hotel-and-casino/photos/square-pool-north-luxor-hotel-casino-v169561/#
http://www.youtube.com/watch?v=PcwXRHHnV8Y
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http://www.youtube.com/watch?v=bFtjG45-Udk http://www.youtube.com/watch?v=OWu0tH5RC2M
http://math123xyz.com/Nav/Algebra/Polynomials_Products_Practice.php
http://worksheets.tutorvista.com/special-products-of-polynomials-worksheet.html#
http://www.media.pearson.com.au/schools/cw/au_sch_bull_gm12_1/dnd/2_spec.html.
http://www.wikisori.org/index.php/Binomial_cube
http://www.kickgasclub.org/?attachment_id949
http://howardnightingale.com/fencing-lot-lines-and-municipal-requirements-some-real-estate-advise-from-poet-robert-frost/
http://www.sulit.com.ph/index.php/view+classifieds/id/6268534/CONDO+MID+RISE+Paranaque+near+Airport
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